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Les diagrammes suivants illustrent la m^thode. 1 * t J 32 X 6 ^ ■' -i \ i ''^*^F %^- -*■? ^;^*'"'<♦' I., »iVt'»r \ *^?' .''■■<*'?>Vii^*^^ y-k 'j< \ 4 .'■■ ■*^' ■#• ./^ .»*»■', u \y ii D. i ,n. I I i ^»»Jt i i^ -J."* y.i PHOTOGRAPHIC SURVEYIJiG »WTf I. 1 'i .fffOTii iMPMIE IraYEYffli INCLUDING THE ELEMENTS OF Descwptive Geometry AND Peuspeotive BY SURVEYOR GEMERAL §F CAJN[ADA . ; '^w^^X r>^ i^ - • % r ' lilTHDDRAPHEn AT THE SuRVEY IIfFICE 1889 • rt •■f -'■v^v- :..; -,..r.:.., ^„^,_. ;_ .„...^,.: ' .i ■■.A ■airtiiwfcWMiirgft \'6if{ -Cs' ^^' ^ fc /" 4f \ --} !»;*, -%,t -v r PREFACE Wh6n the' surveys or Dominion Lands were extend- ed to the Rocky Mountains reglon.it was found that, the. methods hitherto employed were inadequate^l^e <# .y r operations ^ the prairies consisted. merely in de- 'fining the boundaries of the townships and sections: these lines fom a network ove,^ the land by means of Which the topographical features. always scarce In the^lrl^s.are sufficiently well determined for general purposes. In passing^to the mountains, the condl't Ions are ehtirely differentrthe^ topographical features are wp marked and numerous, and the survey of the sec- tion lines la aliiays difficult. often impressible and -^ in mosV cases useless. Th,e proper administration of the country required a, tolerably accurate maprmeans . had to be fx,und of executing it rapidly and at a r i ■^^iH^stj '^iw "!•■. J a* A*!'^'' ■ '."Jf"^ 'i rFTt:7?tri • ' ^ !\" '® PRRPACB '^ - . ' moderate coat. . ^ - » . , The _ ordinary methods of topoi^phlcal SMrveyihg^ i were too alow and expensive for the purpo8e;rapld .^ <»^ surveys baaed on a trlLngulition and on ske^^chea wei-e tried ana proved inaf factual, than pht)tography Vaa re- sorted to and the reWta haVe^^een all that could' be ^ N desired. ^ : » The appHcatlqn of photography ta surveying is. aa Old aa the art Itaelf. Aragp.ln presenting DagueV- ro's diapovery, pointed out tta application to survey- ing.but it waa npt until twenty. five years later that Laussedat gave in the "llimorial de 1 'Off icier du ^ Oenie- a full exposition of the 'method. Kia work wa^ so complete that little has been added L it since. In Oermany, the principal exponent of the system has been an archlte=t.Meydenbau.^:hl, investigation. we^e eontinnea by Dorgens, stolie.Vogel, Jordan and- others. i .. In Italy, the eeletrated Engineer Porro, to ,ho»v. so nanj remarkable inventions are due„as «be first to give hie attention to the proee.arhls ideas .ere «no.ed by other, and ultimat'ely brought out the 9yin«^ Id^ be ^ " >m- PRKPACB . " . ~ 7 Ora„.„.e pbotograp^c eurvey. or the pr.s.Ht .,y. i, M.Jor 8,„er., Ann, bale .,,„.„, p„„^j nire.tcr o^ the Oeographica. -U Itary.InstUute.is aue the credit , Of. initiating. these eurv.ye:th.ir execution .as en- - ^ trusted to Bnginee^L.Plo i.aganlni..„th a etaff of • . able^asals.tanta. The.ork of the Institute i, very remarkable and deserves careful st^. . . ' Notwithstanding the number of thoae^.ho have • : ' written on tfie method, the (rreat artv.„». > o great .aijvant^ss assigned ' tojt and the numerous experimental iunrsys execut- ;ed.there are Vt two example, of its actual ,„^iiy-. ■»nt for ^raotleal p„,„ose,:tho It'aHah and the _^^ Oanadian^Su^veys. ^n P™nce. whereat originated, it ' h.s been completely abandoned, at least ostenaibjy: ' the -Ge^ans use it' for maUng plana ,f buildings. f°=- Which it -is admij-abiy ada'pted, but their topo- ' graphical surveys have "been merely experimental. y i-aiiur. 1 s due to several^ causes .' 1„ thi ' first Place, the scales employ.^ 1/5000 to VlOOO. ■ are too- large. On these scales.such objects as ta- ' iJles and chai ra in a room could be shown on the still no one would think plan, of taking photographs of a a 2 '^ ® PREPACK roon, for making a plan of it. smaller scale, are not generally rehired In Europe. beca,ee good ™aps pn these scales are already in existence. On the other hand, the possibilities of. the «e- ,thod have been over-estimatedat has been asserted that it would apply to almost any country; in reality there are but tCo. classes of surveys to which It Is .eu adapted, and they are the surveys In . mountain- ous country' and the secret surveys. The' Italian and Canadian Surveys are of the fonder description: the latter are In all probability e:ctenslvely practised, although little is heard about them. The authors ,ho have written on the subject had in View surveys on a lar^e scale. executed with great precision, the Canadian Surveys ar'e ,uite different: they are on a small scale and the rapidity of exe- <=utl^>n is such that the more precise processes are not always available. The object of these notes is to Show the amount of infon»atlon which can be ex- tra*t-ed from a photograph »,der various clrcumstan- eas and th, numerous processes at the disposal of i "tfte suirveybr. The^aourc es of photography in that -k i PREFACE • - 1 ■ ■ 9 rospoct,«.o uno,„alle..by W other «urveyln. method. Consiaeratle spaoe has been aevoted to perspect- ive i„.,tn..ent.;a3 no. constructed, they are useless for purposes of topographical ' surveying, but there Is Preclse.avi that Is required being .ore perfect .„r.- »anshlp. .i^architecturaa, surveys they ^y probably be employed with advantage. ** In order to d^nonstrato .ore completely the me- thod, a fo. explanations are given on secret and bal- loon surve,in«:althou.h the subject is ^'L practic- M interest for Canadian surveyors.a knowledge of everything pertaining to Photographic surveying can- not fail to prove useful to those engaged on photo- graphic surveys. A con^iete description of all the instruments invented and of the Investigation, of the various authors being given in the excellent wor. „f .,.„,. Henry A.Reed.u.s.A. -Photography applied to survey- ' ingMt has not been thought necessary to go again _gyer tha aanm grorm dv — — -^ The Canadian Photographic Survey. ,ere co™.„c- ^^^^iiUtiJisitfVA's,... ,V(,i..:, . : ^°' PREFACE Bi in 1887 and cover no, oyer one thousand aquare miles: they are by far the meet extensive ever a,ade. ?or this reason, if for no other, It Is hoped that the exposition of their mode of exeentlon .lu prove ao- ceptable to those interest.d in the development of the science of surveying. While thls^was being printed three new books on the same subject were published in Italy.France and Gennanyrthey are a great advance in the treatment of the subject and show that the method is slowly but surely gaining ground among practical men. 1 have received much valuable assistance from Messrs 0. J.Klotz,Dominio|i.„,Topographical Surveyor and W.F.King, Chief Inspector of Surveys who kindly under- took to revise the proofs. 1 am indebted to them for »any corrections and important changes in the text. ..i I* E.Devilie. ..i '^ -.f BIBLIOGRAPHY^ OP PHOTOGRAPHIC SURVEYING II Pollteenleo.Vols. X and Xl.mimo. 1 'i Application de la photographle a la topographie Ml- litaire.E. Pate, 1862. Bulletin de la Society de Geographie de 1862. Paris, Dec. Ui»orlal de VOfNcier d» ge„le.Nos.l7 and aa.ise* and 1874. ^^ PhotographJgches Apchlv. Sept. 1865. Application de la photographie aux leve'a militalres. A. Jouart,1866. ^ Zeitschrlft fiir Bauwesen, 1867. Archlv fiir di. Offlrlere de. K.preass. Artlllerle «„d Ingenleur Oorpa.Jahrg.Sa.Band 68,1868. la photographie applt^utfe a»x etude. g* \AiL, i-A J. ^ ■tt\! 12 BIBLIOGRAPHY OP PHOTOGRAPHIC SURVEYING. ^ La photographie dans lea armees, Alfred Hanot, 1875. Zeltschrlft fiir Vermeasungswesen, H.2, B.V. 1876, H.23 & 24,3.16,1887. " Journal fiir die relne und angewandte Mathematik.B.95. ' La photographie appllqude au lever des plans. J. Borneo que, 1886. Das Licht,S.T.Steln.Photogrammetrie V.Stolze.Heft 5 1887. La photographie sans objectlf R.Colaon, 1887. * . Photography applied to surveying. Lieut Henry A. Reed, U.S.A. 1888. Les Mondea (Paris) La Nature ( f aria ) La Revue d'Artillerie. (Paris) Bulletin de la Society Prancaise de Photographie (Paris) / RECENTLY PUBLISHED La fototopografla in Italia.per Luigi Pio Pa^anini Ingognore dell 'latitutc geografico militare, 1889. Les levers photogi-aphiques at la photographie en voyage, par le Dr.Gustave Le Bon 1889. Die Pho^ogramotrie oder^^3^^33^^^^ ^^ ^^^ pt.,g. •A Kopp6.1889. PHOTOORAPHIC SURVEYING CHAPTER I. DESCRIPTIVE GEOMETRY 1. DEFINITIONS. PLANES OP PROJECTION. The .olyject Of descriptive geometry is to represent bodies and to solve problems 021 flares in space by means of their projections on certain planed called " Planes ' of projection ? 2. GROUND LINE. For this purpose, two planes inter- secting each other are employed : they divide space into four solid angles. Usually, one of the planes is vertical and the other one horizontal; their line of intersection is called -Ground line- and is denoted by the letters XY. =A^ raSPRESENTATlON OP A POINTTTel XAY,.Fii:T7^ the vertical plane, XBY the horizontal or ground \v^-', v.!^l-lWi& J)'#iW«:'4 ft' »:';i ^* PESCRIPTIVB (JEOUBTRY plane and P a point in apape. Fr«n P.draw the 'per- pendiculars P/?, Vp' to the groiind arid vertical planes;/; is the hor- izontal projection of the point P ^"'^ P' its vertical projection. ^^^•^ ^e* the vesical plane be re-i volved round the line XY aa an axis, until it coin- cides with the ground plane;the point p' will fall at a point p' such that the line pp- will be per- pendicular to XY^ For let a plane be drawn through Vp and P/.^';it is perpendicular to the ground plane as containing Pp and perpendicular to the vertical plane as con- taining P^^ ;but when a plane is perpendicular to two other planea.it is perpendicular to ^their intersec- tiorv. therefore the plane pTp^ u perpendicular to XY, ^nd its traces o^ and op^ on the ground and ver- tical planes, are also perpendicular to XY, since a line perpendicular to a plane Is perpendicular to all the lines passing through its foot in the plane. But op' being perpendicular to XY. ..> niust also. be perpendicular to XY;it follows that ^op^ i^ ^ ♦^ REPRESENTATION OP A POINT . i « straight line perpendicular to XY.^ The ground plan. .111 then be as eho.n in Fig.2, ,,„- being the distance Of the point P from the ground plane and o^ ita dia- tance fron. the vertical plane; both points ^ and/>' \ are on the aame perpendicular to the ground line.ae V " ■ ■ . ' \ // explained above. X -.c Fiff.g if It is usual to i-opresent points In apace^by capital letters, the horizontal projections by italic letters and the vertical projec- tions by the same italic letters accented. It has been shown that the two projections of a point are on a perpWicular to the ground line. Inversely.any two points on a perpendicular to the ground li^e are the projection, of a point Of spa.e. Por let the part of Plg.2 above the grLnd line be revolved round the line XY until its pile be vertical as in Fig. i. Through^ draw a parallel to . - op^ and through^' . parallel to .^ : they win „eet ih a point P. B«1^ -^ i» perpefiaicniir to XY by hypoL.la »<» It 1, .i„ perpendicular to o^, .Inc. J,-!, the ^6 DESCRIPTIVE GEOMETRY angle of the vertical and ground planes; therefore o;t>' is perpendicular to the ground plane, because it is per- pendicular to two lines in that plane. It follows that />?> parallel to «?X is perpendicular to the same plane. In the same manner^it .may be shown that /pT la perpendicular to the vertical plane :- Therefore /? and />' are the horizontal and ver- tical projections of the point P. ///< nj-/,jj rirj rsj ft) (7) rsj r?) \ fly' '-rtt' " I >>. «f a ^ <^ ' Y Pig. 3 Pig. 3 shows the representation of a point in various positions. (1) is a point in front of the vertical plane and above the ground plane. (2) ts in front of the vertical plane and below the ground plane. " iJ^lJjiJ^i^Jfche plane and below the- I •^1 ground plane. •f w,a ^ I » .y , d, t^' / * re op' is per- ls that } plane. ' la ver- id the I ' STRAIGHT LINB 17 (4) is behind the vertical plane and above the groxand plane. .... (5) is in the ground plane in front of the ground line. (6) is also in the ground plane linoc but behind the ground (7) is in the vertical plane above the ground line. (8) is also in the vertical plane but below the ground line. ^ (9) is on the ground line. 4- REPRESENTATION OP A STRAIGHT LINE. If perpen- diculars be drawn to a plane from every point of a straight line, the locus of the feet of the perpen- diculars is a straight line and is the orthogonal ' projection of the first oae. The projection of a straight line may also be defined as the intersection of one of the planes of projection by a second plane perpendicular to the first one and containing the given line This second plane is called the projecting plane. .A-s^m^ght line yt perfectly- defi ned/^y its projections, because it is the intersection of the. B \ ^ ■^ \ DBSCRIPTIVE GEOMETRY two projectihg planes. There is however an excep- tion when the given line is contained In a plane perpendicular to the ground line; the two proTecting planes (Coincide and the projections of the line are not sufficient to define it:the traces must be given. The "traces- of a line are the poi^its where it intersects the planes o^ prpjection. These points are easily found by noting that the vertical trace, (Pig.4) being in thej vertical plane, its horizontal projection must be on the ground line. but it is also on. the horizontal projection, a^ , of the given line,, therefore.it must be at the intersection of the lat- ^, J- ter with the ground line at o. By erecting at o a perpendicu- lar to the ground line, the vert- ical trace will be found at c. Similarly, the horizontal trace is obtained by erecting at e^' a perpendicular .^^ to the ground line; ^ is the horizontal trace. . I n A straight line may occupy valrioua positions with reference to the planes of projection; theqa positions are illustrated below, / X <>/' Pig. 5 V 2 _ f * ^i«»5 shows a line intersecting 4^ „ the vertical plane at ?', above the - ground line and the groxand plane at <^^»in front of the vertical plane. Pig. 6* The vertical trace, ^', is / .-■ y below the ground line; the horizontal =twic B.WTiw-w^T¥^rwT^r^r^^^ti^ \ Plg.e of the line between the traces is in the lower front dihedral angle = ^tiM^^'. ^ fit- .'ij.w4»^- .. tC' < . ^^ ^*.i. »..*•-» J ^ B«S f &■ 20 Pig..-^ X Pig. Pig. 9 Pig«10 DIISCRIPTIVE GEOMETRY Pig. 7 The^"iertl6al - " ^ trace, /^' is below the " ground line;the horizontal trace, a: , la behind it. #. tS'*^ Pig. 8. The v*rt. trace, ,/^', la above the grouhd ' - line; the, horizbntal trace, ■ / ^r STRAIGHT LINE 21 to the groirnd plane, therefore Its intersection with ' the vertical plane. «// is parallel to the iround line: i- Pig.llo any line In a plane parpen- I ' xv^ V "^ dlcular to th^ ground line, Tbe horlaon- , tal and vertical projections o%ncide K^. Pig. 11 tie . I "% and are on a perpendicular to the ground line. -As explained above, line in this case is no^ defined by its; project which do not change. whatever may be the direction ^ the line in the projecting plane, but when the traow are given, the line la defined. Plgia^>.^.t,i.^.' - * .". . *■. 23 % • I- >'•-. '^ -^ STRAIGHT LINE and both projections Intersect ther-^^- When a line is in the ground p]ane,its horizon- tal projection is the line itself and its vertical projection is the grovuid line. When a line is in the vertical plane, its vert- ical projection is the line itself and its horizon- tal projection is the ground linoo The grour.dline is its own horizontal projection and its own vertical projectiono 5o TH#JGH A GIVEN POINT, DRAW A PARALLEL TO A GIVEN •LiNE. When two lines are parallel, their projections Of same denomination' are also parallel, because their projecting planes being perpendicular to the same plane of projection and passing through parallel lines, are themselves parallel to each other and therefore th^ir intersections with the plane of projection are parallel lines. It follows that wrten a parallel to a line r^^, «^'*;Pig.ie has to be drawn through a point * CrXi'y it la auf fie lent to draw — \'« Pig. 16 through c a parallel to aS and b 2 ■1 -f. Us 2* DESCRIPTIVE GEOMETRY, through c' a parallel to dh' then od , r^'is parallel to ai, a'b'. ■ When two lines ^, cr/l>';cd^c'd'.mz,xrj) intersect % each other, the points of in- tersection^ and /?', of the projections are .on the same (I h perpendicular to the ground^, \ : linec It has been shown ^n 1]: 3 that this is necessary in order Pig. 17 that /> and ^^ may represent a L- point in spaceo It follows that when the points p and y?^ are not on the same perpendicular to the ground line, the lines a/>, a.^y rrl,c'4^o not intersect, that is to say they are not contained in one plane. 6o REPRESENTATION OP A PLANE « A plane is repre sented by its traces on the planes of projection, that is to say by its intersections with the said planes. These traces meet in a point <^., Pig .18. of the ground line.which is the point where the plane outs itc The vertical trace of the plane is ./P', the hor- ~i* ont ttl trac e 1 a a:P ■ ' ■ — ~ — J^;^^^i:kia.^Mifx' ^UI^'joALi^l J^'ti^ . ^^M^ilMa)k.^'^^ft.\' el ect $ 3 •der PLANES ' 25 When the plane is vertical, its trace ocp'.Pig.ig on the vertical plane "x^ Fig. 18 ^ V /' Fig. 19 X — Y P\ jP4g,ao= being the intersection of ^ two vertical planes, Is a vertical line. u But a vertical line is perpendicular to the ground plane and to all lines con- tained in this plane by which it is inter- sected: therefore it ts perpendicular to the ground line. It may be shown in the same way that the horizontal trace aP, Pig. 20, of a plane pferpendicular to the vertical plane* is a --" me* p6i*ji4iiig.23, are also parallel to it. A plane P«^P', Pig. 24, per- pendicular to the ground line, has its traces perpendicular Fig. 2-4 to -it. The ground line being '. ;'--."r„4a-.'- V Planes ~ 27 perpendicular to the plane. i^ perpendicular to all the lin^s passing through a. In that plane and there- fore is perpendicular to the traces aP. aV,', Two parallel planes have their traces parallel, because the traces are then the intersections of two parallel planes -by a third one. 7. LINE CONTAINED IN A PLANE. A line contained in a plana, has its traces on the traces of the plane, since any point of the planes of projection not on the traces is outside of the given plane. Hence fol- lows an easy method to find a line contained in a plane P P'. Pig. 25, when one of its projections a^ , is given. The point «^ where the ^' horizontal projection of the line intersects the horizon- • tal trace ct ? of the plane, is the trace of the line. Its vertical projection a.' is a point of the vertical projection of the line. But J|he pgj ■x« Pig. 25 ^nters^otionr^or^^^ With the ground line Is the projection of a point of the line AB 28 DESCRIPTIVE GEOB£ETRY contained in the 'vertical plane, that is, the projec- tion of the* vertical trace of AB;then if at -^ a per- pendicular hp' be erected to XY,it8 intersection J>' with aP' wiJLl be the vertical trace of AB and the vertical projection Hfill be obtained by joining aJ^h' 8. POINT IN A PLANE. When a point M is contained in a' plane Pa;p;pig.26,one of the projections, my, of the point is sufficient to define it. .V- '•-- Pig. 26 • To find the other projec- tion, wy, of M,let a horizontal line be drawn through M in the plane Pig.28 is drawn as explained above from any point 6f the REVOLVING PLANES 31 given line and then a plane is drawn through the two lines by joining the traces of same denomination of the lines < ' When it is required to draw through a given point a plane perpendicular to a given line, perpendiculars to the projections of the line are 4rawn from any point of the ground line: they represent the traces of a plane perpendicular to the given line and there remains only to draw a plane parallel to the first one and passing through the given point as explained in § 9« . 11. REVOLVING A PLANE UPON ONE OP THE PLANES OP PRO- JECTION, Por making constructions in a plane other than one of the projection planes, it is often conve- nient to revolve the plane round one of its traces upon the ground, or vertical planes; the construction is then effected and if necessary, the plane is re- volved back to its original position. The problem can always be reduced to finding the position of a point M of a plane Poa'',Pig.29, after this plane has been brought into coincidence wttn 0T« or the planes of projection, the^gro^d plane for instance. by a revolution round its horizontal -*^ ■• • f 32 trace ' ocp. DESCRIPTIVE GEOMETRY '; •• Prom the point rrv, draw a perpendicular mK to ^he trace aP of the* piane and join MX and Mm. the plane MKm is perpendicular to .the ground plane as containing M//|; hence it .contains the ,||rtical line / X /n T'-tiX - .A' M,. at K to which.' ctK is per- pendicularc But cxYi is also, ^. ^y construction, perpendicu-^ '^B.T"^%m^ therefore it is Pig. 29 perpendicular to the plane , • .w-KM and to KM which is In this planeo Consequently when the plane is" revolved round its trace.M will fall on a perpendicular KM, , to aP. * Let us, suppose now that the triangle MK;/2/be re- volved round' K.. on to the ground plane; the angle y^nvyL being a right angle, the side mVi will fall in mm, parallel to ^P; ^...^whlch is the height of M above the horizontal plane, is equal to ;>^;it is therefore easy to oonstruet the triangle «n^ by . taking KM, equal to K/aj^ the position of M, is^ f to ia -ne ' Iso, CU-, is ne ; In ?d I. re- in obtainedc 'REVOLVING .PLANES 1 < 33 The constmction lines on the figure have for object merely to show that m,n^ is made equal tohm', KH equal to Km^ and that the point m^Arif Uga in, the plane PaP' ( § 8)0 , ; ' \ - A similar construction would be employed to re- volve a plane upon the vertical .plane^ \ It may be observed that the angle m^K^n; la the angle of the given plane with the ground plane. The position of a line rovblved upon the hor- , izontal plane is determined by finding the positions of two Of its points; its traces for instance. Let ^* oJf,a^, Pig.ao.fee the iine and Po-P' the^ plane. Prom If f ' . ' ' ' draw a perpendicular *K to »■ - ■ • < «P. The same demonstrate. on as in the case of Pig. 29 will show that aP is per- pendicular to the line K6' in space, therefore ^*in / / • ""^^ ■ ■ '/ 1 >., • V- 1, ■ ^ig.30- e; Its revolution round afP will fall on K&. produced, and KB^ will be eoual to^ V —» :|| %^, '* DESCRIPTIVE GEOMETRY K*'. But K*' is the hypothenose of the right angle triangle, Kl.i; ,hic„ can be constructed at Kti,j by. -akingK^ equa^toK?^ th, position of 'b. 1. Ibt.in- ed. No, it must be Observed that K fro- ■»uced;^^ 1. the revolved line.^ul a z the re vol...^^ trace of the plane PapC Vo REVOLVING- PJANES 55 To revolve a plane back into its original po- aition, inverse constructions are employed. Let it be required for instance, to find the projections of the.poiht ^ ,PiR.29,wh§ji the plane Fr^ is revolved back to Pap'.. The angle of P«P' with the ground plane is first determined by the construction f?iven above: ' then from M^ a perpendicular is drawn to « P and at the point of intersection K.ap angle //^K/z/^ia con- atructed equal to the inclination of the" given plane on the ground plane. Knif is taken equal to KM and from y;v\a perpendicular m^nv is drawn to M, /^;;7#^i8 the horizontal projection of M.and nvm, its height above the ground plane, from which the vertical pro- jection Is easily found. A line is, revolved back into its original po- sition by repeating in inverse order the consti»uc-- tions given for revolving it upon the projection plane. Let «^B, ,Pig.30,be the linetfrom B, and ,> rtl? I REVOLVING PLANES 37 perpendiculars aa;, and bb' are drawn from «' and b to the ground Une; .O, and ,.'i^'are the projections Of the line b A when revolved back to its original place. A perpendicular to .< P is „ext dram from M, : its intersection with at gives the horizontal pro- jection m Of the point M:the vertical projection is Obtained by drawing through ,«. a perpendicular to the ground line and producing to its intersection «^' With ah' instead of the line aj., .',■ e. parallel to the vertical plane may be employed. Let /^-.; Figc32. be the point, PaP' the given plane and Puij the same plane revolved upon the ground plane. From /A' draw a perpendicular /7^ n. to «P; the point M, will fair on this line. Then through >V?;;Kr; draw a^,a^d; parallel to III ■ h' lii. b Fig. 32 the vertical trace .P' of the plan, ( } s ). »,an R,P' «P'and as its trace <^, doe. not move, the line .ill * f •<•• 38 DESCRIPTIVE GEOMETRY fall in «^B, , parallel to jtI^ so the point M, will be on a^B, ,but this point is also on nvn , therefore it ia at the intersection M, of mn and «^B, . To find the projections of the point M of space when it Is given revolved on the ground plane in M, , draw through M, a parallel a.^^ to the trace c* P^ and a perpendicular M, /?^ to o,Po Through a draw a2} par- allel to the ground line; it is the horizontal projec- tion of a line parallel to the vertical plane and passing through the point 11. But the horizontal pro- jection my of M is also on the line M,///, therefore'^ it is at the intersection of M,w/ and al? , The vertical projection ni> of 11 is on the per- pendicular rrvm^ drawn through rw to the ground line; it is also on the vertical projection a^h' of the parallel to the vertical plane, which is obtained by drawing from ay a perpendicular a^' to XY and through aJ a parallel to the trace a P^ The inter- section of a2>' and /t^^.^ gives the vertical projec- tion W^Vf If. The oo'^at'^otlons are stin ^''^^^r simp lified r When the given plane is perpendicular to one of the ^fc r REVOLVING PLANES 39 planes of projection. Let PaP',Pig.33,be a plane perpendicular to the grotmd plane Sindzn^/n^ a point of the plane. The point M in space is on the vertical line passing through /«^, which line is in the plane PaP' and is ' perpendicular to the horizontal line 3tP. Therefore, when BxP' is f.f. [^'J'f X _•^;_.^^..J._..,^y ■>■>" x/' M, Fig. 33^ revolved round aP.the line ^iH will still remain perpendicular to aP and the point M will fall in M^ at a distance mU, from aP equal to the height of M above the ground plane. But this height is hrn^'/ therefore to determine the point M, , draw at m/ a perpendicular to aP and take m^U^ equal to ?7y?n' . ) Instead of revolving the plane round aP»it may be revolved round ofp'on the vertical plane. The point M will then describe in space an arc of circle of which the vertical projection is the line /^'M par- •t ■ allel to XY and the horizontal projection an arc of tji rclr 77?^, with or as a C9im>r Sfiff a;^it7 as radlu^ 't When the plane PotP' coincides with the vertical plane, c 2 9^ !&-a^s»i.:.'.:M^:. . ■.^ 40 . DESCRIPTIVE GEOMETRY the point M of the plane rrrust be somewherb on the line //t'M^ qnd its horizontal projection is t. Then if a perpendicular to XY be erected at t and pro- » duced to its intersection M^ with m/M.^ , m/ will be '' », ' . the required point c K%. X To find tne projections of the point M whose position M^ revolved on the ground plane is given, draw from M^ a perpendicular V^m to otP and fromTTiy a perpendicular W//Z/ to XY;take ,^w.'equal to //i U^ , height of the point M above the ground- plane; m.m' are the projections of the point. ^ The projections of M^ are found by drawing through M, a parallel U^r to XY, which is the Vert- ical projection of the arc of circle described by M, when revolved back to i^s original position; take • '^my equal to the distance 1*,v of M, from the trace <*P' and through w draw the perpendicular //^//t'to XY rn/my are the projections of the point. 12. INTERSECTION OF TWO PLANES. Let Pa P' and Q/Q', Fig, 34, be two planes: the points H and N where thft _jtr msea q# th g p lM-wg meot .^re the^ trat&ea of the — — ^ J' /V m .» line of intersection of , the planes. The projections t, \.^ T-' ft. ■ - INTERSECTIONS 4I U?i yfriyn of the intersection are foimd as explained in * 5 4 by letting /. fall the per- ' / ^ pendicuJars UmJ «'' and N/^ to the ground line and. ^^S-34 " joining M/z,/^'N. l3o THE' INTERSECTING PLANES ARE BOTH PARALLEL TO THB GROUND LINE a Let PQ,pV,RS,RS'.Pig.a5,be the traces, of two intersecting planes parallel ^o the groun(J linerthe construction given in § 12 does not apply and recourse must be had to an auxiliary plane. Draw a plane TOT' perpendicular to the , ^, , ground line. The. line of inter- section of the two given planes is parallel to the ground line and so are its projections. If the projections my and /7i/ of the point M where this line inter- sects the plane TOT'were known, A' J*' n a. Q_ y' \^"' 'fi , :■! 9 Fig. 35 rtfte proJectlonB of th6 Tine i t is elf would be obtained at onee by drawing through rtT/ and 77z> parallels to the ,i' ^. 42 ground line.. DESCRIPTIVE GEOMETRY To obtain M,let us revolve TOT' around OT upon the ground planerthe intersection ^of tOT'andPQ p'q'* ' •of which Ijhe traces are c arid d', will fall in ^D • OD, being equal to 0*^^. Similarly the intersection Of TOT'and RS R's'wlH fall in oM, .OB, taing equal to 0^' and the point M will come in M, .at the inter- section of ^D, and r^B, . Prom M, ' draw a perpendicu- lar to OT;the point of intersection w/is the hor- izontal projectiorx of M. Th^ vertical p5r(ijection is. obtained by making Qm^ equal to fwM^ .this, being th<=. height .of M aljove. the ground plane, 'ihen through m^ and rrv^ draw the parallels ef.ef, to the ground line; they are the projections of the line or intersection. 14. THE INTERSECTINO PLANES CUT THE GROUND LINE AT THE SAME POINT. Let PaP'.Qc^Q'.Plg.se.be two intersect^- ing planes cutting the ground line at or ,Draw a plane TOT' perpendicular to this line; oris a point of the line of intersection of the plane^and if the pro- jections my »nd/^^ of the point M where this line cuts the plane tOlr'were known, th e jppo .lee 11 omurviL_^4^^ 1 ir^tersection would be obtained by joining c;r//2^ and a /t^'. «" I T Mj^i t- /' is obtained by making 0/^' equal to m\K, , Then draw . - 'f' VJ ■11 "?!&' iO JJ Y Fig, 38 44 DESCRIPTIVB GEOMETRY projection a parallel q^r to x P. In the case of a plana parallel to the vertical plane { Fig. 38 ), the ,/''' *; horizontal projection of the intersection Is the trace QR of the vertical plane. The * vertical projection is a parallel i/W to the vertical trace a p' of the other plane. .16. PLANES PERPENDICULAR TO ONE OF THE PLANES OP PROJECTION. When the two planes are both perpen- > dicular to one of the planes df projection, their Intersection ia also perpendicular tp this plane and its projection on it is the point where the traces of the planes meet. The projection on the other plane Is a perpendicular to the ground line passing "through the above point. 17« INTERSECTION OP A LINE AND A PLANE. To find the intersection of k line and a plane, another line intersecting the first^one is drawn in ttaep lanp ; t v^g ^ ii point required is the intersection of the two li nes. I 2 ^i&A*: li- 'Cia.i-ii^ 'I ■4i ',1 n^ ! /'/' " : 4 V INTERSECTIONS 45 Let ..v., ^7.; Pig. 39, be the line and P„p' the ^lane. For auxiliary line. the intorsec t Ion of PvP' by one of the pro- jecting planes of the given line. /t.fi^y, for instance, may be era- ployed. " ^ V ^° obtain the projections Of this intersection, draw the ^ig'39 perpendicular ^A' and cr' to * the ground Une and join o7,"; a.o'^'is the inter- , section. It meets the line ^^,^7.' at //^/.i.' which is the point where the line cuts the plane P^p'. 18. INTERSECTION OP THREE PLANES. The intersection Of three planes may be found either by constructing ' the line of intersection of two of the planes and then detennining the point where this line cuts the ' third plane or by constructing the lines of inter- section-of one ^of the planes with each of the others: the point where the two lines meet is the point of Intersection of the three planes. ^^ THROUGH A POINT.TTDRAirASmiGHT-LI^^.SfcS': WILL MEET TWO GIVEN LINES. To draw through a point/ *^ DESCRIPTIVE GEOMETRY a straight line which will meet two given lines not in the same plane. a plane is passed through the point and one of the lineB» The point where the second line pierces the plane is ascertained (§17) and by join- ing this point Of Intersection to the given point. the line required is obtained. 20. DISTANCE OP TWO POINTS. Let e^^^/,^;pig.4o,be two points; to obtain their distance, one of the pro- jecting planes of the line AB may be revolved about its trace upon the corresponding projection plane. Let us revolve, for instance g' KBa^ around j a^. The point A will fall inj^ on a perpendicu- lar a^\ to a7^, the line ak being the height of A above the ground plane, that is the distance r^^: Similarly B will fall ih B, , on a perpendicular 2f^ to aJf , and at a distance from i equal to sy . The required distance of the polihts is A B . ^ ^^<» « o»B tract io,^ may be somewhat simplified by .^^, "A l'ig.40 ob«rvln« that- If . im. b. dra« through ,. parallel 1 1 DISTANCES 47 to iV B, .its length a o is equal to A B, ; therefore Instead of constructing the trapezoid .^A, B/^ , it will be sufficient to erect a perpendicular to ah at 7. and to lay off on it a dis^tance Tjr equal to the difference between sb' and /V . ^ 21. TO LAY OPP A GIVEN LENGTH ON A LINE. T^ie con- struction given in 5 20 may be employed for laying off a given length on a line^AB ( Pig. 40 )^J^rt^ the / projecting plane on the line ad as an axis and lay off the required length A, D, on A, B, . Then revolve the projecting plane back to its natural position: the horizontal projection of D, will be at the plane t ^ DESCRIPTIVE GEOAIETRY N '1^- i>V '"'> ■^ Pig. 41 containing the two parallel lines. Revolt this plane around its trace ^y <', until ' • ' if ' n 1* coincides with the ground plane ( 5 11 ). Let /Z.B, and M, be the revolved positions of r//j' and M. Prom M^ let fall a perpendicular U, K to /'i^B, ; it la the distance required. 23. DISTANCE PROM A POINT TO A PLANE. The distance from a point to a plane may be obtained by dropping a perpeodicular from the point to the plane ( 5 10 ), . finding the point where it pierces the plane ( § 17 ) and determining the distance of the two points. It is more convenient to pass through the point a plane perjfendicular to one of the traces of the given plane. This auxiliary plane, being perpendicular to the other one, contains the perpendicular from the point to the given planerby revolving it around its trace upon one of the planes of projection, the follow- ■ ing almple construction gives at once the solute n. .-p I 'S. the problema 4\' 'k -^ DISTANCES 49 'V x Let RirP' , Pig. 42, be the plane and m/ft^ the point. Throiigh f/irn^ pass the plane (^Q,' perpendicular t0 6«p'and '' revolve fi aroxind y^Q upon '- the ground plane. The point *, A describes the arc of circle AA ,and BA, is the ( /. ■'' ^ intersection of the two planes revolved upoji the Fig* 42 ground plsme. The point M is on a parallel to ^Q passing through-^'. In revolving the atixiliary plane, Wde-, scribes the arc of circle y»'^ and the line z^^'m falls, in cu^ .still parallel to ^Q. The point M remaining during the revolution of the plane at a constant dis- tance from the^ertical, plane. will fall on a'parallel to the ground line passing through ^n^j thei'efore U will come at''' the intersection of o}^ and /^l^ -There remains only to let fall a perpendicular from M, to :.. BA^ :it is the distance required. , ^ 4u .__DlSTAIl^ QF TWO TA RAUMh FM W BSr ' The dfs tance of two paft'allel planes may be obtained byl interj ting D flU^/^U^kt/^.^ A^^iiflW j^ t^^iltj^'\^$v. iifja 'n? 50 DESCRIPTIVE GEOMETRY them by a third plane perpendicular to both and re- volving it upon one of the planes of projection. Let Paf?',^Q,' Pigo43 X '3/ "C ». + '-^k be the parallel planes. Draw a plane ROR' perpen- dicular to the vertical traces and revolve it upon the ground plane around OR as an axis. The points R' and s' de- 4 " ' *^ig.43 scribe the arcs of circle RI^ ,Vs, :the lines RR^ and S^ being the intersections of the given planes by the auxiliary one. These lines are parallel and their distance is the distance of the planes. 25. DISTANCE OP TWO STRAIGHT LINES. Let AB and CD ;.Pigr44 be two straight lines not contained in one plane; it is required to find their shortest distance c This distpce is the perpendicular to both lines. Through any point of AB, A for instance, draw a par- allel AP to CD and from a point of CD, let fall a perpendicular GH on the ■ p lana ti av Ti mA.-„i- ^i -n ^wx^»i.u.^vui.»i uji ^«t Tine pj.an«h SJUr.^ Through the foot"—" of QH in. the plane BAP, draw a parallel HK to AP and I i Jiif ti* nVti-.t-fibr"^-^ ANGLES 51 through K anatlier parallel KM to HG. the lino KM is / perpendicular to both Al /\ G D II '/', \U Pig. 44 lines* Although present- ing no difficulty the construction requires many lines and Is om- itted here* 260 ANGLE OP A LINE WITH THE PLANES OP PROJECTION. Let^lt be required to find the angles formed by the line ah,a:i'. Pig. 45, with the planes of projection. The angle of the line, with the ground plane is the same as with the line a6 , since the plane 7j'm^ is perpendicular to the grotmd plane This angle can be obtained by revolving the triangle y^a^ around i'b as an axis upon the vert- ical plane. The vertex a. describes the arc of A^ ,^ x X ^-^' •n' d Pig.4'5 ^^^^ ^ ^^^^^^^ ^^ "the angle at c being the angle of the line with the ground plane. D 2 i [fs/^ii^JAwi, Wzsj «> it 1^ ^ - ■''" — ' •' Si. m.:.. ^^ DESCRIPTIVE GEOMETRY Similarly the angle with the vertical plane is obtained by revolving the triangle «^v^ upon the ground plane aroimd «<^' as an axis. The vertex 7^' comes Ih d, the angle «./^' teing the angle of the ^ line with the vertical plane. . * When the line is contained in a plane perpen- dicular to the ground line, such as ^^; Fig.4e,the ' angles are^found by revolving the plane upon one of the planes of projection, the ground plane for instance: the vertical, trace, ^'describes the arc of circle ^'B, and the re- volved position of the line is «^B^ ; a-&nd B, are the angles with the ground and vertical 5'ig«46 planes respectively. In the cKe of a line parallel to one of the planes of projection, the angle Of the line with the other plane is the angle of its projection with the ground line^ \ 27^Am^or TWO MNgs,, T, find th. aagia^ f p^a^ C- B, '«, by two intersecting lines, their plane is revolved ! i ..fe i.:. ^•i 4 ANGLES about its trace upon one of the planes of projection. Let ^ _flamfl,^ afl^4ii^gj i^^ 4»r fa^ i^~i n ve rtffd . The g t ven^ ling- tg the intersection of the two planes: the triangle ohd . ?, _,w. rj8h'- 31. ANGLE OF A LI^ WITH A PLANE. The angle of a line with a plane is the conqplement of the ahgle of the line with a perpendicular to the plane. So in order to find the first angle, a perpendicular may be erected to tl^e plane through a point of the given line ( % 10); the angle of the two lines is then 'de- termined ( § 27). * 32. METHOD OP ROTATIONS, The method of rotations is a process employed in Descriptive Geometry for facil- itating the solution of problems. It consists in ro- tating the whole system of the projections or only part of it, around an axis perpendicular to one of 1 the planes of projection,until the system assumes a position favourable to the solution of the problem. 38. ROTATION QP A PWN!^, ^t it be required to ro- tate a point //«^//2^7 Fig. 50, through an angle oo, around + a vertical axis «/, a/b'.tti^ projection //a^ will describe As- .' u-iif.-'t'Lii-A :-zlyk'.':*'.&i..\L=u .■-^, ' f RpTATIONS gg an arc of circle ^^;^ with center at a. and subtending an angle equal to to . But the point M, during its motion re- mains at the same distance frdn the ground plane; there-' fore its vertical projection, * rnf, travels on a parallel -nVm^ to the ground line. So when the point 7??/ has described the % / / nv m^ Am, " > a ll/n..-'- Fig.Sp arc^o.,: W point m. is in r^t^;, .t the intersection of the perpendicular to the ground line through^ with the parallel to the same line through m; . 34. ROTATION OF A LINE. Let ^^, ^'.; pig.51, be a ' straight line to be rotated around a vertical axis ^,^'^ until parallel to the vertical plane. Prom c/ , let fall^the perpendicular orn/ on €tb^ and rotate the projecting plane containing «^. around 1«te axis. The point 772/ Will describe an 'r ''ig.51 «rc-of^circW and stop at I t^e pen|mdicular to m^ i / ''1 60 DESCRIPTIVE GEOMETRY the grornid line drawn throngh r . The projecting plane t will then be parallel to the vertical plane and so will the lines ob and AB. The new poafttion of r^Jf, is Obtained by drawing through /^e^, a parallel ^/; *, to the ground line and making ^,^ . ^^„z. , ^^„^. . *^. In their motion around the axis. the height above the /gjround plane of the points A and B of the given line d/ to the ground line. But cv', , the newyertiSal projection of A.must be on the perpen- dicular through <^v to the ground line and since it is also- on the ground line, it must be at their inter- section in ^/.Similarly. /?./ must be on the perpen- dicular b'^J to the ground line and also on the parallel <^?V , therefore it must be at their intersec- ■^ion ^/. The rotated vertical projection is then «^'/$; 35. ROTATION OP A PLANE. A plane may be rotated by turning three of its points, not on a straight line ( § 33), or a point and straight line, both in the plane, or two of its lines ( $ 34). T he following ma. I / ♦ thod is a simple one ^' , I' i / » ROTATIONS Qi PaP,Pigo 52, is a plahe to be rotated until per- pendicular to the vertical plane, about a vertical axis of which the horizontal trace is at «r From c let fall a perpendicular- ckT on op and rotate aP until c^ is parallel to the ground line: 9? will .then be perpendi*cular to XY. It is the rotated horizontal trace of the plane. ' , Now draw any horizontal line g}v,^'k>^xn the plane , ^ ' P^P; produce c■■•;;■- V .mt 62- . DESCRIPTIVE UKOMETRY the parallel ^'h/ to XY.and will stop at the. inter- section of rj'h' and y,//^ produced, nine e^,/^/ ia perpendicular to XY. ^' The rotated line y,/^y/ , is still parallel to the ground plane and is now contained in a plane per- pendicular to the vertical plane: therefore it ia it- self perpendicular to the vertical plane- Its vert- ical projection is the point ^; , which is also its trace and consequently a point, of the vertical Trace of the- rotated plane. But or, is another point of the new vertical trace, therefore the rotated plane is ij ^,'^:^ ' I' The angle ffi^.^l is the inclination of the ro- -ated plane on the ground planelthis inclination' is the same before and after rotation. The plane might now be brought parallel to the ground plane by a second rotation about an axis per- pendicular to the vertical plane. 36. DISTANCE OF TWO POINTS. As an application of this method, the determination of the distance of two "joints may be given. Im^X awjdd, Pi^o53,be th^. points o Rotate the i J '^ I 1 t <% i J ^ Pig. 53 SPHBRICAL TRIANGLES 63 vertical projecting planu containing a «nd J around the vertical line through a until it is parallel to , the vertical plane. The pp.^nt ^ describee an arc .of circ^Le ^^ and stops at ^ on the parallel r^ to XY; ^'moves on a parcel to XY and stops ai J/ at the intersection of the parallel Vh: and the perpendicular ^^' ^ to the ground line. The rotated line is now parallel to the vertical plane;it is there- fore equal to its vertical projection «.^'V- The in- cunation of the line on the grour^d plane is that of the vertical proje<|^ on the ground line. Another solution of this problem is given in^ 20. a7„ SOLUTION OF SPHERICAL TRIANGLES. A sphericaj tri- angle may be assimilated to a trihedral angle byi sup^| posing the vertex of the angle to be at the center of the sphere. The sides of the spherical triangle ^e then subtended by the plane angles of the faces !of - the t rih ed ral a n g and^-the-anglea- of -the triangle are^ the same as the dihedral angles of the trihedral .J angle. X'-- 64 DESCRIPTIVE GEOMETRY . , As usualithe sides of the spherical triangle are designated by cr.^^ and o, the opposite angles being A.B.and C. as. GIVEN THREE SIDES TO FIND THE ANGLES. The three sides of the triangle correspond to *he three faces of the trihedral angle » Develop them on the ground planet placing one of /' the edges, OQ,, Pig. 54, perpendicular to the ground line and re- volving the feces* -fz^ X / . y i . // »»f4r .•*■ n Pig. 54 and o about the edges OQ and OR, upon the ground plane. The in- tersection of the trfhedral angle by the vertical plane forms a pyraiRid of which is thet vertex and OQR one of the faces in its natural posi^tion. Since OQ is perpendicular to the vertical plane, the planes of the two faces interHbecting along OQ are also per- pendicular to the vertical plane, therefore OQP,is one th© f aoQB of the pyramid » revolved upon "fche^round ^ plane about OQ, and OP is the third edge of the 'W^ i SPHERICAL TRIANGLES . 65 pyramid, the. vertical trace of which is on the arc of circle described from Q as a center with Q^ as ra- dius. 1 #1) The third edge of the pyramid is also shown in OP, which must be taken equal to 0^ ; IJ ,like P, ,is the vertical trace of the third edge OP revolved upon the ground plane. Let now the face o be revolved back to its natural position, by turning it about OR: the horizontal projection of P, will move on the per- perpendicular ^w^let fall from P, on OR, and when P, comes to its original place in the vertical plane, its horizontal projection will have moved along P,m/ up to its intersection p with the ground line. The vertical trace P will therefore be on the perpendicu- lar pV to the groxmd line, but being also on the arc of circle P P,it is at their intersection. Having now obtained the trace P of the edge OP on the vertical plane, the dihedral angle C is foxind at onqe i*i PqR, since both faces are perdendicular to the vertical plane. _ — L-^ae n eral ly^^nly on e ang l» %» r e quired; in maki nfr — the construction, the edge corresponding to this angl* B 66 DESCRIPTIVE SEOMETRY is iplaced perpondicular to the grovmd line. Should the other angles be wanted, A could be ob- tained from the triangle //-wp revolved around P/? on the vertical plane; Pm^/^ is the angle A of the spher- ical triangle. B is constructed as explained in 5 29 or by any other method. 59. GIVEN TWO SIDES AND THE INCLUDED ANGLE, TO FIND THE REMAINING SIDE AND ANGLES. Let cu, h and C,be given; required c, A and B. Place the intersection of cu and h in 0Q,Pig.54r perpendicular to XY, and the face T? on the ground plane; draw QP making the angle PQY equal to C: QP is the vertical trace of the face a/. Make the angle Q,0^ equal to «> : QO]^ is the face €(y of the trihedral angle revolved about OQ on the ground plane < Taking QP equal to Qi» , the point P is the vertical trace of the third edge of the trihedral angle. To obtain o, let fall from P and p the perpen- diculars Yp and pm ^o XY and OR respectively- Re- volve about ?p on the vertical plane the triangle ^o«B»d-^ te by P;^ and ptn^v^^p-^ ' ^ t h« angle ^ Then produce prrc and take m\ equal to /^P:joi V- » n \ SPHERICAL TRIANGLES " ©7 OP, : ROP^ is c. B is obtained as explained in §^38 40. GIVEN TWO ANGLES AND THE SIDE OPPOSITE ONE OP •THEM, TO FIND THE REMAI^NG SIDES AND ANGLE. Let ^/^A,and B be gil^en: required C, I* and c . i^ Place the face c on the groiond plane and the intersection of ^ and c in OP, Pig, 55, perpendicular to the ground line. Through P draw Pq making with XY «htf angle B; PQ is the vertical trace of the face «>, singe a. and c are both perpendicular to the vert- ical pilne. Draw OQ, making the angle a. with OP; P Oft '; is the ffice a of the trihedral angle revolved upon the ground plane about OP as un axis.Q^ is the re- volved vertical trace of thp edge ^1^ Making^ then PQ equal to PQ^ givep the trace Q. Through Q, draw Q nv. making the angle A with XY^and let fall the perpendicular Q<^ /^ '"I *-"i 'o L .' ■*:* X ■'^' -^V'- ii- < "4 'f> . til- - O ^ vertical trace. Q /6t the edge opposite to <^ is on a circle described from P as a center with PQ,^ as radius . Throu«^ P pasg a plane perpend I' "j^*' k'-" Pig. 56 dicular to OR: its 'w" \' Ok SPHERICAL TRIANGLES 69 horizontal l^ce i^B a line "Pw perpendicular to OR and its vertical trace the perpendicular PS to XY. The intersections of thfs plane with the two planes of projection and the plane of the face o, form in space a triangle SPw in which P is a righ/angle an^/w/ is* the angle B. Revol^ng this triangle about /^P upon the vertical plane, in SPt^^ , the point S is obtained. But S is a point of the vertical .plane of projection and is also a point of the plane of the face oy , therefore it is a point of the trace of the last plane. Joining then RS.the intersection of this line with the circle Q^ Q is the vertical trace of the edge of the trihedral angle opposite \o ,c, of the other triang^, and the sides « , ^ , o, ; l%» w. DESCRIPTIVE GEOMETRY "^,,.^v.., "Of the 'lupplemel^ary triangle are tl^l&t^l^tff^r' the angles ^B.C^ of the o|her|aneo- fe^^-i* ^^■ ■ Fronl*:any po|t 0.p|^, l%e- i^^rl^^^ . - ^||. .'t!*!!^ triti^drat^nglevletr';;^^ ■»!^'* '* .,Pig»ij7 faeei?. The anglQ of OT ai^ OV is the sup™ plem^^ of the angle of therplan^a to which they are perpendioiaar. \'- ■ But the angle of the It' * planes is the ang^e B of the trihedral angle ; there- fore TOY or ^ is equal to ISQo ■- B. :■''.' ' " '-. " • ■ ' Similarly/: TOSa <% =180o- tJ v'^-.- '■■■■•■■ ..■■■' The plane TOY containing perpendiculars to a> and e, is perpendicular to .both; the pendicular tj^heir ihtersectio A^ / Dft is perp lar to TOY. For erpetadicular to TOS and DP angle of DQ and DR or a^, ia the^ au '.'*» 'Y' SPHERlCAt TRIANGLES / . 7,1 angle formed by VOT and TOS or A^ | ^ , A = 180O- (I . In the same manner, it may, be shown that: and c, = 180 "-i^, ' ' , . Hence the trihedral angle 9TSV is the suppiemenr- tar'y angle of DP(iR. ' , ' • ' ' A >43o REDUC;PION OP' AN ANGIB TO THE HORIZON » The re- duction of an angl^to the horizon is an application of the solution of spherical triangles » Ihei^ an angle is observed between two points, which, are not in the horizontal plane of the observer, the observed angle requires a correction to reduce it to the angle formed by the projections of the points on the ground plane. Por^hat purpose the observer measures the angular elevations or depressions of theipoints. H,|«^ 'j,K^ tee as vertical plane 6f projection the'plahe palsing through the observer and one of the points. Assume aV points,^, yifr58^as' the place of observation f and dr^.throu^'^it the li|nea /A and PB, making with ' the^grormr-rtrrr anires equal to the ele^ti depressions ^ kfid^^^^f a ^nd B. ons or e 2 "x J 72 DESCRIPTIVE GEOMETRY TH&iines PA,PB and the vertical. ?/> form a tri- J ^' : 'V ^^ A r f/f,~:3 ~Y B\\ Fig. 58 hedral angle in «rhioh the faces are-»©0<*-a, 90°-^;^^ and the observed angle. A pyramid is cut off this trihedral angle by the ground, plane, the base of the pyramid being the triangle pAB,in which pA and pB are two sides and p the observed angle reduced to the horizon. The thi^d side may be found by revolving the face APR of the pyramid U around AP.upon the vertical plane- Thif*face wLll come in APR, ^ the angle at P being the observed angle and" /^.. PB, = PR =: PR^ We have now the third side of the triangle A R :hence describing arcs of circles fVom p and . A as centers with A% and PB, as/i^adiuj? rispectis^e- ly, their intersection is the point R and A^R is the ** required aQ glp.. _ - ____ «^ i 'WV CHAPTBR II PBRSPEeTIVB «^ 44, Perspective la that part Tf ^etry #hioh treats , , Of the representation By figures WL on i surface of objects placed beyond it. Oene rally this surface is a vertical plane; it ^s called " picture plane ? The fig* was drawn on it, according to the rules of perspect- iva.produce on the eye as faxgi^Mom is coiicemedV^ the same Impression a|S the objects themselves seen in their actual places <> i Suppose a transparent plane surface, such| as glass, placed between the eye and the objects to be represented. .If the outlines of the objects seen ^TOroiigh the glass could be trabed on it, the 1 thus formed ||uld be an ea^ct perspective ^* ^.. % 74 Cons PERSPECTIVE aual rfclj^ from the eye to a point .M .Z>:>i Of spa^Hfrray pierces tha^lct^^e plane in a se- cond point, which is called the " perspective " of the first one. .0$^m^ ,^. ^.. The visual rays from the eye to all the pointf«^ fa straight- line form a plane those intersection ,^*|th the picture plane is the perspective of the * line. Consequently, the perspective of a straight J.-.; line fs another strai^t line. When the line is a curve, the visua^ays to. it^ % variout .poUits form a conic surface whose vertex i's at the eye and whose intersection with the. picture plane is thb per^eckjLve of ^e curve. A surface of the same ' Of the objeoei« the *n^^»tic^ o^hi f sUi^ac e by the picture pl^iie « W^"^" > .*? . r^'^8 fonned by the visua> rays tangent to the ^i«^b^^tline gf an oiyictrL perspective ^'1 %^.0 DBFINi:^0|]S. -yie Aground plan- i. the hori- % 2ontal projection of the objects to be represented; vthus for the perspective of a landscape, the ground plan is the topog raph ical jlan of the ground; for a building, it is the horizontal or ground plan of the ^ ■'i. % \ '% ^• -:MM. :^imM « N /'- DEFINITIONS 75 building ( ABCD, Pi^r. 59 ). The "gro und plane " is the plane on which the ground ^lan is placed ( KX5Y,Fig.59 ). For a lands- cape, it may be, f 01* instance, the horizontal plane paaa- ing through the datum point of the topographical plan r ^"d for a building, the bisement or first^ floor plane. Any horizontal plane may, however, be used aa ground plane, provided its altitude be taken Into accountrthe ground plan does not chahge, whatever the altitude may be. The " elevation * is the vertical projection of an object: the elevations of a building are those plans of the building which show the front, reax', or sides. The "picture plane". as already explained, i« the plane on which the persip^vo is drawh ( FFXY.Fig.SQ ). Generally,.it is vertic^fc placed between the eye and the object to be represented, but none of these rules is absolute. Perspectives are sometimes drawn on planes which are not vertical and objects are re- presented which are between the picture plane and the' ©yia.1 Such- a-^pos4^i©n oft objects is the rule and riot the exception, in perspectives used for surveying. J..-. f ''O PERSPBCTIVE i*»en they are taken as representations not of the ground itself. but of a model of it reduced to the scale of the map. This convention will be found further on. Objects are even represented which are behind the observer, the origin of light, for in- stance in the eonstruotion of shadows, but this is merely a geometrical conception to which the usual definition of a perspective does not apply. The " ground line * is the Intersection of the ground and picture planes ( XY,Pig.59). The -station- is the point supposed to be oc- cupied by the eye of the observer. fS,Fig.59) The "foot / A^ -\s ■ o "^'-^ J- ^^iM «f _ Y ^P^' Pig.SO Of the station " is the point where the vert- ical of the station pierces the ground plane ( S , Pig. 59). The -principal point - is the foot of the perpen- dicular drawn from the station to the picture plane; ■& DEFINITIONS 77 it is shown in P, Pig. 59.. * ' fl^e "distance line " is the line between the station and the principal point ( SP, Pig. 59 y, Ita length ia the diatanqe from the station or from the foot of the st?&tion to the picture plane. The " horizon plane_» is the horizontal' plane passing through the station. It contains the dis- ■« tance line and cuts the picture plane on a horizon- tal line passing through the principal point and call- ed "Horizon line" ( HH,Pig:59 ). The distance between the horizon line or the principal point and the ground line is equal to the altitude of the station. The " principal plan e" is the vertical plane per- pendicular to the picture plane and passing through the station ( SNQ, Pig. 59). It contains the foot of the station, the principal point and the distance line, le " princip al line" is the intersection of the Ipal and picture planes ( QN, Pig. 59 ). It is perpendicular to the ground and horizon lines and /■ o intersects the latter at the principal point, A. "front jjlane " is^ a plane^ parallel -to th©. picture plane / ! ^rj,J -'^a PERSPECTIVE i\ "front line" i plane, there Tore any 11 ane // .N V/V // Y ♦ ^ -Pig. 60 any line contained in a front n* parallel to the picture Tn Fig.60,.th9£e points, lin-^s and planes are repre- sented by- th€ir ortfaogonal projections: the ground plane is taken for hbrlzontal plane and the picture plane^an-i for vertical planer ^5.' in . the station, .$» the foot of tha station, .v' or P the principal point HH' the hori- - ST- zon line, sp, s^ the distance line and ^/> N the ^rin- ,\ ' CTpal plahe. " '^ 46 rERSPEGTIVE 05- A -P0I41T IN ifes GROUND PLANEo .Let ,% XYs-. Jji-^iGl, be the ground plane, ^YN the picture plane, S th0 station and M a point 5a .* ,\» "^ M X *4-; in the ground plane^ The " . '■*? . ^* perspective of M c^n the « picture plane iai the point where the s|raight line SM ^, LINE IN GROUrJD PLANE 79 • horizontal trace. Thus we Have the first relation ' between a point of the ground plane and its porspect >,,, , • ■ ■ , ■'_ i" . . . • ■ . Ive.they are llhe traces of Ijhe visual ray on the ground an^. picture planes respectively. ■ '^ . ' . " - ■ ' Fig^62 represents in ? ■ - ■■ * '. / *' orthogonal projection the ''construction of Pigo,61; .y.y^ ^' is the 8tation,M the' point of the ground plane j-«, w/.ff the j^ visual ray and ff uthe -' fc^ (*■ X ::Pigo62 perspective of M. The points .i ^, 11 and ^v are the traces of .y«r, T/t'.s*/ 47. PERSPECTIVE OP A LINE IN"- THE GROaNI>^IiANEo It has been shown in §« 43 that the perspective of a straight. line 1« the/intersection with the picture * plane of the i^lane contain- -•" ing the 4^ation anifl th^^ given ^line* '- ' ^ \fc // ,\ •*' *^- ^ ^ Draw'a^plane through ^^ the 8traig» line AB ftnd ' - r. ^f." Si V- '•-1 ^ ii5--6* the s ta ti of^ %>i g . Qii\ ■■ The interse^jon g^ of this (1^' •^.si: t>. 'M. < f. 80 ■r ^i PERSPECTIVE plane with the picture plane is the perspective of ^ AB. Thus we have this relation between a straight line in the ground plane and its perspective: they are the . traces on the ground and picture planes, of the plane - containing the station and the line itself. . . In orthogonal projection^ / p *^® ^'^"® being in the gi•o^lnd plane, the horizontal projec- tion is the line itself, AB, Pig. 64, the vertical projec- tion Is thb ground line. To draw a plane through "the station ^T,ana the aine AB. draw through ^'P a par- allel to AB;^the horizontal projection is a parallel to AB through s, and the vertical projection a par- allel through P to the ground line. The vertical trace is at *^', the intersection of c? with the per- pendicular cry to the ground line. The horizontal trace of the plane conjainlns s¥ and AB is the line y Pig. 64 AB itself, since it is in the rrofend plane. The vert- ica] trace passes through r', t ■he trace of the line .yr P^-. which 18 contained in the plane; it must ai ) >\ ' / 30 pas-s y a... ) / >\ ' 1 LINE NOT IN GROUND PLANE 81, through A, therefore the vertical trace is the line * Ac". Hence Ai> is the perspective of AB. 48. PERSPECTIVE OF A POINT NOT IN THE GROUND PLANE. The construction given in § 45 ciyes riot change, when ; the poir^t to be plaqed. in ■ ■« ,- «' • fi perspective is not in .t^§^ • ' drawn through the station 4 »r - fit:- •/»*' a/ ^P, Pig. 65 and the point Fig. 65 , . ■ . \ ■ nuTV- The vertical trace; ^ , .• ' is the perspective- of /^z./7i>.- The horizoA.^1 |racy«//?> on the groimd pla^ne; hence it may be stated as a general rule, that 'the .per- spectlves of a point on,:iho f, round a-nd picture planes are the traces of tne line joining tfie stition to the point 49. PERSPECTIVE OF A ilNE NOT IN THE GROUND P^NE. Let ah, a^h', Pig ae, be a line not 'in the grou^d»plane : to obtain its perspective a plane oiust Jae passed' through the station- ^P and the line ^^, a^B'j the interaction of this plane fith t^ie jsicture planJ '-^ -yrf '& .,-. .'..Jl-t-. IV- !»»■■'■ 'i '^Mfi'^^^'^~'^'-''^^^^^'^^^^^-&'\i¥t"-iVr,'T 1. <1; 1-: 82 PERSPECTIVE tha/t is the vertical trace of the plane, is the per- spectiye of the ^- \ ■ : ■ ' i ^ "~-v\ /c Pig. 66 line. " ' , Throxi^h ^P, 'ft draw a parallel sa^/ Pd' to ^, a'^'j both lines are con- tained in the plane ■ *o be drawn, there- r fore the traces of the plane are the lines ^^ d'^' ' .^ . •' , . ■ ^ ... -C joining the traces of same denomination of the par-, allela;and 4'*; the vertical trace of the'pl^ne.is ^ the perspective of a^^ cuB'. Let us now consider another line, ^ ^/^ parallel to ^, ^'^;. the plane pa8f?ing through ef,e^' a^nflth^ station, ^P,mu3t again contain the parallel sd^, 7^< : through the station; therefore the vertical trace of the plane. which is the perspective of <^, C is the' line f'd; joining the veirtical traces of the two i^ar- allels. Hence the perspective of any line parallel \t) «^,«>'r.will p^a through the poirfC .^'. This result . .^ could be foreseenrbec^sa, irhen^a ^y«St^m^f parallels ""'^ \ ■■•.'^- . ».-._. i. ill, A. i^* -i«i*a:^t ..'7^' .. ? i iI>Cd.:V'--;^tJ4:j^--^-. V" 1% Ty -'.'«: .;:-:.,'. vaJjishing point 83 / \ .♦•■• A ^ > .^ .. 'K . •*: ■ • ■'■■: t-A »'■.■•, i'^;' ■*■ has to be claced in per8pectjLve»all the planes serv- ing to praject them on the'picture plane have a com- mon line of intersection, parallel to the general di- rection of the system and passing through the station. ■ ■ t Its trace on the picture plane must therefore be the Bemmon point of intersection of the perspectives. This point is called the " Vanishing point * of the parallel lines, becauslPit represents the parts of the lines which are at infinity; the perspective ends or vanishes fit that point.. . ' ^ Th€ horizohtaf traces of the planes are the -perspectives of the parallel lines orr'« the ground plane* Like the perspectives of the picture plane. "^ they all meet in a common point, which is the hori- zontal, trace pf the parallel line through the station: it is tHB .vanishing point of the perspectives of the ' grotind plane. Therefore- it is. s^en that when a p: is drawn througft the station and a line^in sp&ct^Bie nes ^ traces of the'planfe on the picture and jgrpund pl^ne^ are the" perspectives of-^ the line on those planes. • 50. POSITIONS OP. THE VANISHING POINT. A horizontal line hais its vanishing po,int on the horizon line ^ ^ '■ F 2 4' "• 1, ">v>*vy ■■■■> * , t \^ Ki .i't ■-" '-'A Sii'^'^^ ■': 4^ ■ .,1 . 1-, / \ - 84 PERSPECTIVE because the parallel drawn through the staiti on, being horizohtal,is all contained in the horifon plane and has its vertical trace on the horizon line. Perpendiculars to the picture plane being par- allel to the distance line' have for vanishing point thA^rtieal trace of the distance line whiph is the principal point of the perspective. ^^' ■ ^f' \ * The vanishing points of Ijorizontal lines making an angle of 45° with the distance line are called ^distance points* ( d'.D Pig. 67 ):their distance from the principal point is equal to the distance line, because a horizontal line inclined at 45° with SP, forms, an isosceles triangle SPD in which SP = PD. Lines in the principal ■ \ ■ Plggje have their vanishing point on the principal line* Two of these lines form angles of 45^ with the distance line, one above, and the other below the horizon. Their vanishing points are known as •uppei: *2l ^o^er distance joints ■; they are also at the same distance from the principal point as the \ \ /J JO Pig. 67 ^ \ VANISHING LINE , ' • ,, Ho ,• . - Lines parallel to the picture plane have no van- ishing point. It will be shown later on that their perspectives are parallel to the lines themselves and " ., ' • ' do not meet. ; • . . - P ' 1. VANISHING LINE. Through the station ^P, Fig. 6a, . pass a plane TVZ parallel to a given plane QRM. The vert- ical trace VZ contains the V traces of all the Tinea drawi/'through the station parallel to ^RMi-it is there- fore the locus o5 '^he van- \ .N Pig. 68 ■0. ishing points of parallels to the plane QRM. This trace VZ will he called the " vanishing line " of the plane ORM or of any other plane parallel to it (1), (1) The term "va-nlshing line" ia usually applied to the perspectives of parallel lines: admitting that the expression "vanishing point" ia a proper one, the line VZ cannot be called otherwise than "vanishing line". The terra is used here with that acceptation only. #. ■ '»- y :"-*^ ; 'I t i 1 86 PBRSPECTIVE The horizontal trace VT ig in l^M manner the vanishing line pf the perspective's . 52. LINES OR FIGURES IN FRONT PLANES. The perspect- iv^ of a straight line contained in a front plane is another straight line parallel to the first one. For the plane containing the station and the given 1^; being cut by t»o parallel planes, the picture and fi»ont planes, the intersections arejarallel lines. But these Intersections are the Une, Itself and its perspective. therefore the perspective is parallel to the given line. • \ ' Let S, Fig. 69 be the station, PP', FP/ the picture afld fror\ • planes and ABCD a \ polygon in the front /plane. J^in SA, SB, ^ ■ . sol SD; these lines ''ig'69 intersect the picture plane-^at ^^^ o^ r^/ the polygon «^o 'v -■?•■■* ^ > tW ■i I > ■i I PIGin^S IN FRONT PTJVNES Q,i pyMmd of which the polygon of^ the front plane is the base^ and the pe^specti'y^e^ a section by a plane n^ paral,lel to the base. -It is shown in Geometry that „ when a pyramid is cut by a plane parallel to the base. the section is a figure similar to the base* The front plane being parallel to the picture plane, the per- <, spective must be similar to the origina?. figure. ^ ^ % ■ ■ . " ^'^■•'. / "' It follows that a curve in the front plane is , ■ , i " ■ represented by a similar curve in perspective, because " '' , ■ such a line Q-kn be assimilated to a polygon with a great number 6f aides. ■ I When the front plane is beyond the picture plane. as in Fig.69,the p^spective is smaller than the or-, ,.■■% iginal' ffgurejit is larger when the front plane is between the statiorl and •tho picture plane, but in either gase it is an exs^ct representation of the figuLT-e itself, on a different scale. This ipcale.or the proportion bfeween the perspective and! th« or- ■"■**fe=»„. igihal figure r is called the " scale of the front plane * . ^ *,. it is the\|)Vo£ortion of the distance- line to" the di4- t^nce betwe< strai^t ^!^n« parallel to the picture ^lan». is 4^ e station and' the front' plane-. f 2 . 88 ^^>* : PERSPECTIVE cq^t^ined in a front plane and isl|roaented in per- •specuve by a line parallel to itself . therefore par- allel lines Which are al.o' parallel .1^ the picture plane have parallel lines for perspectives and have no vanishing point. The parallel to the given lines pas^ng through the ^stat.on, h.ing parallel to the picture plane has no' trace or, it. Vertical lines are parallel to the picture plane and appear in perspective as parallels to the prin-. cipal ^ne. ^rstal lines parallel to the picture p;ian^ rspective parallel to the horizon line 53. M3^.\SURING LINES AND MEASURING POINT* ^-'ig.70.be the picture plane. S the'^tation and AB a > i^re ■!» i \ V *- •■ -t^ 90, PERSPECTIVK f M — eat.. - ..uax to ., . o„ ,, -.^^ ^^^^^^^ point D, Join to M, and call J the ..♦ the intersection with VA. The line seen In i^ < m Ai is „uai to AD. therefore the part seen m^S i- e,»al to q D. The line AD is called th« "measuring line- of AV, because it serves to measure the length of the line in space corresponding to ar^ portion Of its perspective AV; M i3 the " measuri ng point ", V« -a, not dr.™ in an. particular direction, therefore th. direction of th. „easurin« line.paranel to VM. i, indete^inate. It i. ^.,, ,, ^, ,^ ^^^^^^^ ^^ ^^^ horizon line. The position Of the .e.surin* point depends only on the vanishing point, therefore th. s^., ^asuring point ^y serve for all line, parallel to th. ..« direction. the .... „„„„,^ ii««^r„^. ,„ .^^ ^^^^_ Fig. 71 % SCALE 91 having th«lr Yertical traces on it. Should the line VM be drawn parallel to the vertical trace of a plane, this trace would be a measuring line for all lines contained in the plane. If the measuring line is taken parallel to the" horizon, th« measuring pbint of any horizontal line i« on tba horizon line.sin^e the vanishing pbint is on that line. All lines in the same horizontal plane have then for measuring line the vertical trace of the plane, and lines In the ground plane have the gpound line. There is no measuring line or point for lines In a front plane, because they have no vertical trace or vanishing point; the scale of the irgnt plane has to be employed when the length of such a line is wanted. The distance points are measuring points for lines parallel to the distance line. 54. REDUCTION OP A PERSPECTIVE TO SCALE. Hitherto, it has been assumed that in the constructions, the real dimensions of the figures were eiq>loyed. It would be quite linpracti cable to do so in the % ' .j Bfe^' 92 PERSPECTIVE generality of casea.'the dimenaions must be reduced to a certain scale in order not to- exceed the limits of the paper. . . A changed in the position of the measuring line permits the use of reduced distances. Let V.M.and AC. Pig. 72, be the vanishing and 1,1 measuring points and the measuring iine of the per- spective AV. The part of the line seen in ]^f is equal to BC. Through a point ^ a/ ot AV,draw the paralle C // ^* V * Pig. 72 ojy to AC and let us vti it as a measuring line; the length corresponding' 'to /ir is dc and we have the proportion: '.:* * c^ _ V«/ CB VA Thus the lengths obtained are all reduced in the proportion Of ^. Therefore in order to obtain at once the l^gth.6n a certain scale, of a line seen in perspective.it is sufficient to reduce the distance, betieen.t.,0 measuring line and the vanishing point in Mil i I SCALS 93 * the proportion of the scale to be employed. Thus if Va. be made the one thousandth part of VA,the dis- tances will be obtained on a scale of 1000 •M is the measuring point and* coc/ the measuring line, of a line having V for vanishing point and ct/ for trace on the picture plane ;^he new line is therefore parallel .to the line joining V to the station and to the original* line seen in perspective, but its distance from the station has been reduced to the scale ad«3p«S« Hence, to obtain the length reduced to/Boaie of a line seen ir/'perspective, reduce to scale the dis- tance of the line fropi the station, moving it parallel to itself in the pitohe containing the station. The same conclusion is arrived at in a more di- rect manner otheirwise. A figure ABCD,Pig. 73, forms with the visual rays o fi. 44 ^' 1^ 1 ^^"-\^-^^ .J joining it to the station, a pyramid, the intersection of Which by the picture plane f. • ^ ' Fig„7J is the perspective of^/^. Let the pyramid be ■\ »• \ \ ,J.\ . • < ■ ■ ■ - -n • jllll m^ ^' '" ^/._',i'''i"^.' ' ■ '-y ' ®* ■ •' PERSPECTIVE cut by a plane parallel to the baee ABOD:the inter- ..aection A, B, C, D, is almilar to ABOD. the proportion being |A, rue linee A, B. .9 o, „„3ured by .ean. Of their perspectives oiJ^J^r t. ••••• will therefore be the lines AB. Be reduced to the eoale 5*- 3A The same demonstration applies lo any system of flgures,whenever every point of the system has been moved in a straight line towards the station. so as to reduce its distance from the station in the pro- portion Of the scale given. Hence we deduce the fol- Iqwing important rule:- *o lay off dimensions reduced to scale or to measure them from a perspective, assume that the system fonned by the statidh and the original fig- ures or objects.had been reduced to scale when the perspective was executed. 55. TO PLACE IN PERSPECTIVE A POPT OP THE GROUND PLAlME. I8t. By means Of the principal point and a distance point. Let M.Fi8.74,be the point. XY the ground line.p' and D the principal and distance points. the picture \ V \r i*!.* "Sv CONSTRUCTION OP PERSPECTIVE 95 plane 'being revolved upon the horizontal plane. Draw MA at an angle of 45® and „ MB peiT)endicular to t'he ground line. The perspec- tive of AM is the line AD joining the trace on the picture plane to the dis- . X M 'J9 a '^''■0^>9>- ^>"tance point. The perspeo- ' Fig. 74; :\y%^^.\ ■■ tive of MB vanishing at the princiiral point is PB, therefore the perspective of M is at (, ^1 ray from the station to the point "'"^''"^""^^-*^-^—^ Wiethe projection o^ the perspective of m revolved upon the ground ^ P^ane..oin .Mandate erect a perpendiculars, to the ground line:t.e perspective Of Mis on that ' perpendicular at a distance ^^ e.ual to ^e^/, - CONSTRUCTION OP THE PERSPECTIVE 87 When a great ntonber of points have to be placed in perspective, this last method Is very oonvenient> In practice the perspective is not constructed on the ground plan itself, as the operations would become confusedithe plan and perspective are kept separate. Let ABCD, Pig. 77, be the ground plan, X, ^ the ground line, and s,p, the trace of the principal plane. Join 5, to A, B, C, and D. On the paper which i'^/'> .Ji, '1? / -^ <' 'l.V s ^ '^ -f; ■(> >^ ^' i a.^ is to serve for the per- spective, draw the ground line XY and take a point P as irfl»«ection of the principal plane. Take on the edge of a piece of paper the dis- Pig. 77 tanc 88 from Pj to«..>S., • 1 ■f^ ' ^i , and c »arry them on XY in ^»>/3., /,, •C ; At the last mentioned j >oint8 erect perpendiculars to the ground line. .f A At another place draw a line s^ B^ to represent the intersection 1 of the ground and principal planes: / s:';;.--- 5-'%., G • u ^SajW^^SjC-^^^ >d ^-'-.. - .-iKo ic- ^^sji**^\ - -i- J - - « k.^ 1 -- '■' -■~-W """•^mmtmum i) >'■ ®® PERSPECTIVE place the station S, at its height!^ abgve the ground plane, take s,;y equal to the distance Line and draw the trace of the picture plane, ;?^^ perpendicular to ^^ ^ . ~\ On the Qdge of m pirce of paper. take the dis- " tances of A.B,C,D. from the ground line X, \ and carry them on //.^ .Join S, to A, .B, ,Q ,Q . Again take on the edge of a piece of paper the distances of oc^,A,y,,^,, from/i and 'lay them on the per^ pendiculara a, ex, yd^jS, /^/,4J^. This give.. cx/3y

ID PUN*. When a line is in a horizoniil plane. ^ that pl4ne »ay be taken temporarily as gro^d plane and Changed when the perftpeotlvt, has been obtained-. If in anr other plane, the perapective^may be - found by means of the vanishing point and horizontal trace. The latter is placed in perspective as ex- plained in \ 55.and joined to the vahishing point. For liiies in front planes, one point of 'the lin, ia placed in perspective and through U.a parallel to the line is drawn. 59. THE DISTANCE LINE 'IS AM AXIS OF SYMMETRY OF THE PERSPECTIVE. A perspective is ayiaaetric.al with ref- eT^ence to th6. distance llne.all points of the picture plane at the awne distance from th'e principal point having the same geometrical properties. Therefore any plane perpendicxUar tp the picture plane may be taken as grwJnd plane. or any line through the ' ( ^ ^ CONSTRUCTION OP THE PERSPECXJVE 101 principal point as horizon line. So wnon figvires are- "coyitained i'^n a plane perpendicular to the picture plal-re. their perspectives can be obtained by taking •the p:t.ane of the fi^rei^ for ground plane.', and its '■ " ■ ^ / vver^cal trace for ground \pae. . ^ 60. GIVEN TI^ HEIGHTS OP TWO POINTS AND THEIR PER- SPECTIVES.'TO PIND THE VANISHING POINT AND TB^CE' ON ' PICTURE PLANE OP THE LlN^ JOINING THE GIVEN POINTS. L^t HH and P,Plg,79,be the horizon line and principal point. Jt and yS two pointer of the perspective. Draw ' EP parallel to the horizon • line Mt a distano^ equal to the height of < :it is the • trace of the horizontal plane containing the point of jspace co?i»e8p^ding to a. fhe perspective of thai. ■t ,y aM -K-k'-" ,i^- Pig. 79 'perpendicular to the pictiire plane passing .through ©c '• • . } - '- ' . Is P« :it8 vertical trajj^ -is C.i,Draw CD perpendlc- -ular to the horizon li;rie and equal to .the height of ^ above the plane of a. , D is a point of the- picture plana at the same height as^, and PD Is, the. ^- 1- t>.Vta>,,7'i^. ..Jr■^ 1 ,_.__.. __ „^,_ ■ i< I I'l i i ■ 't; 102 ' PERSPECTIVE . * ' perspective .Of the perpendicular to t^e pioture plane passing through- D)PD is in the same vertical plane aa PC and if p*^oduced will meet the vertical of a seen m perspective at aB. The point of intersection A Is at ^the same height as D and^, therefore A/^ is a horizontal line and its vanishing point is on the horizon line afG. But A;^ ' and «^ are In the same vertical plane having for vertical trace the perpen- dicular GV to the horizon line, therefore the vanish- ing point of ^^ is at its intersection V with GV. To find the trace. draw through D the parallel DL to the horizon line: it is the trace on the picture plane of the horizontal plane containing AG, and the trace of AG Is at its intersection L with DL. But AG and 'I ■ y h-) Let SP,Pig 81 be the distance line, T]d the principal line and FO the intersection of the front pjjane dontaining the vert- ical line, by the principal plane. Mark on PG the in- tersections of the horizon- tal planes, Join to S and produce to tj^ the inter- sections are the projections on the principal plane of the points requiredt In practice, the oonstiniotion is made on the perspective: r|0 Pig.82,being the perspective of the vertical line,NM is taken on the horizon line equal Pig. 81 rW^ ■INTERSECTIONS BY CONTOUR PLANES 1C5 ,n X to the distance line and NQ equal to the diatance of the vertical line from the picture plana. At Q a per- ^ pendicular is erected to Kh' EUid the equidistance sc»le pinned alongside, so that 0, shall correspond to the altitude of the station ^ r. '0 M if Fig. 82 The construction is complet ed as in Fig. 81 Contour planes being equidistant, t^ife divisions <^Pi J^'f of the perspective are equal:it would therefore be sufficient to find the length of one division and to carry it on the perspective of the' vertical line. A^' 62. TO MARK ON THE PERSPECTIVE OF ANY LINE OR CURVE CONTAINED IN A VERTICAL PLANE, THE INTERSECTIONS BY A SERIES OF HORIZONTAL PLANES. Let fX(f, Pig. 83, be the perspective of a line contained in a vertical plane: that plane contains the vertical seen in per- spective at its horizontal project ion, HH' the horizon and XY the ground line. The intersection of the grotind plane by the vertical plane containing the line seen at ', the trace on the picture pllMie of this intersection is at E, where cub produced meets XY. Through the foot of the station s, draw sv parallel to (vb and z^V perpendicular to X^ meeting the horizon line in V. V is the vanishing point of par- allels to a^ . , But the intersections of the horizon- tal planes by the plane of ot^ being parallel to a.h, V is their vanishing point; and since they are all in the same vertical plane, their traces are on the vert- ical ED of the picture plane. Hence the equidistance scale la to be placed along ED, taking care 'that the point B of tho scale corresponds to the altitude of the ground plane; the divisions of the scale are >v« r J 108 PERSPECTIVE Jolneti to the vanishing point and produced to their intercection with the perspective. 65, TO MARK ON TIER PERSPECTIVE THE INTERSECTIONS OP A, PLANE, LINE OR CURVE BY A SERIES OP HORIZONTAL PLANES. The in-tersections of a plane by a series of horizon- tal planes are horizontal lines parallel to the trace on the ground plane, of the plane Intersected; the ^^nlshing point of these lines Is the point of in- tersect ion. of the horizon line by a parallel to this trace, drawn through the station- Let (\ ^, Pig.SS.be the perspective of a line or curve in the plane POP', XY the ground line and HH the horizon line. Through the foot of the station s draw sir , parallel to OP and erect the perpendicular ?^ V to the ground line mfteting the horizon line in V: V is the vanishing point of horizontal lines in the ^ plane POP, and consequently. Pig 85 of the intersections of that plane by the horizontal planes. The traces of these \ / INVERSE PROBLEM OV PERSPECTIVE 109 lines on the picture plane are on OP'and the vert- 's ical distance between ^hem is that of the horizontal planes: therefore place at 0,on a perpendicular to the ground line, the distances of the horizontal planes or the scale of equidis twice, draw parallels to the ground line through the divisions A,B,C of the scale and join A,BiCi to the vanishing point* These lines are the perspectives of the interaecticAs o\ the plan POP' by the horizontal planes. 84. INTERSECTIONS OP A PRISM, PYRAMID OR CONIC SURFACE BY A SERIES OP HORIZONTAL PLANES. The Intersections of a prisiti or pyramid by a series of horizontal planes can be drawn on the perspective by determining the intersections of the edges of the prism or pyramid by the planes and joining the corresponding points' by straight lines A simllair process can be applied to 4 conic sur- face by using g^eneratrices Instead of edges, and also by employing tangents to the Intersections, parallel to the tangents drawn to the curve of the ground plane forming the base of the cone. 65. TO PLACE A POINT OP THE GROUND PLANE BY MEANS OP 110 PBRSPBCTIVE ITS PERSPECTIVE- To restore a figure by means of its perspective is the converse of perspective. Let us consider first the case of a point of the ground plane: its place can be found by inverting any of the constructions given in 5 55. For instance, in Pig. 74, the perspective U) of the point is joined to the principal and distance - V/' points,? and D. At ^ a perpendicular BM is erected to the ground line and at A a line AM is drawn at an angle of 45o with the ground llne.li is the point of the ground plane. „, " * In Pig. 75, join P^ and draw j». A and BM perpen- dicular to the ground line. Join 5 A and produce to the intersection with BM. In Pig, 78, p^ is taken equal to the distance ^m/ of the perspective ft from the ground line and ^ is joined to the station S^ revolved on the grourid plane. The foot of the station, ^ la joined to the foot of the perpendicular ^m^ to the ground line - and the point M is at the intersection of sm/^ pro- duced, by the parallel to the ground line through nv. ' When a number of points have to be placed, the J i- T-'-'^li' , •" -4f 'INVERSE PROBLEM OF PERSPECTIVE m constructions are made as in Fi^.77,but in inverse ' order. The perspective «^// ia given.'the distances P^>'PA^ are carried on X, Y^ ; act, J^^ on ^^. Then s^ is Joined to ac,j5, and on these lines produced the points A,B»C,D.are so placed that their distances from the ground line are equal to 66. TO PLACE A LINE ON THE GROOTD PLANE BY HfffiANS OP ITS PERSPECTIVE The trhce o1 a lino on the picture plane Is the point where its perspective intersects the ground line: the point is common to the perspec- tive and to the line \^, Fig. 86) The vanishing point. V. is the Intersection of the perspective by the horizon line; it gives th* direction of the line of the ground plane. Prom V, let fall the per- pendicular V^ to XY and joiji ys. Through y^, draw, Sk parallel to sv , it Is ' required 14ne> Wi^ a front line a - point of the line is fixed H' V Y \.V Fl^.86 &^^^^^*^t^y^^fSi«i'^'^i J;^:;-/ > > ' 'tlr l: ■ ; /"" i 112 " PRR3PKCTIVE by one or the methods of J 65 and a parallel to the ground Una drawn through the poi^t. ' ^ 67, TO DRAW A FIGURE ON THE GROUND PLANE BY MEANS OP ITS PERSPECTIVE. A figure of the ground plane may bo oonHtructed by me&na of its perspective as described in $ 66, each of the aumnits of the figure > being determined separately. It may also be constructed by determining each of the lineu forming the figure, as in 5 66 . ) kn Irregular figure is enclosed between straight lines and drawn at sight. A convenient method is that known as the " lae thod of squares "^ The ground plane is divided into squares by llnef parallel and perpendicular to the ground line; the network of sq\iares is projected on the per- spective and the figure drawn at sight in the corres- ponding squares. 1 To cjonstruct the perspective of the square9,the \ ■ distanced of IThe parallel lines are marked on the •• ground line in A,B, C,D,E,Pig.87^ the perspectives of the pei^endiculars to the ground line arre obtaiiijed ; - I by joining these points to the principal point P. / / ^ /^ INVERSE PROBIJOl OF PERSPECTIVE lia The principal plane Ja next plotted separately, .'> / V' \ - .1 n '■ /•- r \^ y y^' ,// tf K, li, r/o. -/mi , Fifi.87 Pig. 88 6' K^ being its trace on the ground piano, 3 the station and P P the trace of the " picture plane. Mark the in- Vteraectiona ^ ;g^ ,H, , K,.of«J^ by the lines parallel to the ground line, Join to S, and carry^to Yp the distances of /^ from P ,G ,H#C, -f n through the points so ob- tained, P,G,H,K, draw paral- lels to t})e ground line, wMch w^ll complete the per- ap'octive of the squares. It is not necessary that the sides o^ the aqtiarea be parallel or perpendicular to the ground line./ Any other direction may be adopted, as, for instance, north and south, and east and west In the case H ^ ^ Ififit lJv-A.\ K*- 1, «(rt ^ ■« - ^ X „l 114 Pkrspkgtivt; ^ of t^^'^^rraphical perapQCtlves Tho vanishing points V and V', , Pi/:;. 88, of these lines are found as usual "by drawing throup;h the sta- tion parallels to ttielr directions, to the intersec- " tion with the horizon lino. The points of intersection with tho ground lino ' of the horth and south lines, which will be supposed . to vanish at v, are taken from the ground plan, carried to the groiind line of the perspective, in A,B,C,n,E,F, and Joined to Vifhia gives tn^jjjilferspective of one set of parallel lines. The other set is obtained by a similar process, carrying the points G^ ,H^ ,K ,L^ , from the ground plan to the perspective and joining to the'.'ilpiniahing point % • The squares must be jnnade small enough ■#• th#S(|raught8man accurately, in transferring flMMPTBVre from the.jsejrapoctive to the ground plan. 88. VAWi)^ the groundl Join the foA by pariillelg, to the ground line.J^in these jaoints to Sjthe perspectives of the above parallels; iill be parQllels to the (ground lihe passing through ti^e points of division of //P. Suppose now that the diUanco of a point^of the perspective from the ^ grdund line fle^found equal topni.; then the point of the perspective is on a parallel to the groand line passing through m.. But this - line" is the perspective of ^; a parallel to the ground • line passing through M, there- — " f ■ .^•^ fore the point to^ be found, being on that parallel, is at the distance ;?M fi^om the -ground line, M and >w/ 'correspondingto the sfime divia'ions of the scales - H 2 / •M 'M f im \f9 Pig. 89 V -■fe. ^.,v ». ?*v ?v; -;'.■ It, -, ('I ■-•v J- : f ...y* ,.^fe ^ '"V^ 116. PERSPECTIVE •//P and /'A'.the distance' af the point Is obtained at once by reading the division of /f k corresponding Xoptiv. The scale ponstrdcted as above on pV is called a vanishing scale; when the distance line is constant, the scale is the same for all planes at the same al- > titude below the station. 69. USE OP THE MEASURING LINE. Sometimes the greater part of an irregular figure may be enclosed between two parallel lines, as in Pigi90. A point V is taken on the horizon line such that two lines drawn from it will enclose the figure 5>:!iiJ ^ 118 PERSPECTIVE or hy X, (yj To find the effect on the distance y from the sta- tion to M of an error dx in the perspective, the equation (1) inust be differentiated, considering a> and r/ as variables; this gives: ^ = --^dx - y' dx (2) A I So the error in the position of M caused by an error in the^ perspective in- creases as the square di" y, j^ the distance: therefore the // If it ^ method niust not be employed for points or figures at too great a distance from the station. Pig. 91 The error decreases as the height of the station increases: thus if the height be doubled, the error will be reduced to one half. Hence perspective* in- tended for the reproduction of figures in the ground plane should be taken from as great a height as (N, poaaibl e . The error decreases also as I increases, or as v'-'r.M' 'Ji0!h ', '4'. V-^.m'' INVERSE PROBLEM OP PERSPECTIVE 119 the size Of the perspective increases. 71. TO DETERMINE PROM THE PERSPECTIVE, THE PROJEC- TIONS OP A POINT NOT IN THE GROUNp PLANE, BUT ^QP WHICH THE HEIGHT IS KNOWN, ^he perspective of a point is not sufficient to determine its positioniofher data must be furnished, such as the traces of a plane con- tainlng It, its distance, or its height above the ground plane. . . If the height be known, ^ d***^ a parallel RT,Pig.92 to x^L . 'y *^® ground line representing ■ -in sf: t'^e trace on the picture plane of the horizontal I'ig'92 plane containing the point. The projections of the visual ray Joining the station to the point are sm, P^ ( 5 47 ) lit pierces the horizontal plane RT in W;< and as the point to be found is in that plane and on the line sn., Pf, , u is their point of inter- section, nvnt,. «w cohitpic^oSTsTvo^ always possibl^^^^^^ stance RT may pass through Prthis means that the point h 2 120 PERSPECTIVE is in the horizon plane, in whloh case it cannot be located by means ofl its perspective. Pjx may coincide, or very nearly, with P^, and the construction become impossible or uncertain. The visual ray joining the station to the point is then projected on the principal and grotmd planes instead of of the picture and ground planes: the different steps are precisely the same in both methods. 72. TO CONSTRUCT PROM ITS PERSPECTIVE A FIGURE IN ANY HORIZONTAL PLANE. The methods given in § 65,66, and 67 apply to figures in any horizontal plane, by using the planes of the figures as ground planes; all that is required being to shift the ground line on the perspective to its proper position. '' 73. TO FIND THE TRACES AND VANISHING POINT OP A LINE GIVEN BY ITS HORIZONTAL PROJECTION AND PERSPECTIVE. Before proceeding to consider figures in various planes, it is necessary to show how the plane of a figure and the traces of straight lines can be deter- mined. ( JlAX^^jxd- J mcl fe^ > glg yOS, represent the persp^c ii • tivs and horizontal projeotion of a line. At b erect V Pig.9i5 TRACES 121 a perpendicular to the ground line: the trace of the line on the picture plane \ must be on that perpendi- cular and also on cXjS therefore it is at their . Intersection ^j'. The vanishing point is the trace of a parallel to the line, drawn through the station: the horizontal projection of this par- allel is si^, drawn through the foot of thestation parallel to a l and its trace is on the perpendi- " cular KV to the ground line. But this trace is the vanishing point of acj, therefore it is at the inter- section of /V and ay^ produced. The vertical projection passes" through the trace V and the principal point P;producing it to the inter- section with XY and erecting the perpendicular >;V^ to XY,the trace on the ground plane is found at V . The linfr^joining V, to a( ig the perspective on ^fehe^^romir plane of the givdh Tine f f 49 | whos^ trace is the intersection of ci:y^ and ud . iic.b^iid£.k»'iik^!;^iZ. 122 PERSPECTIVE The trace on *the ground plane may also be found by revolving the projecti-ng plane of nh around its vertical trace ?>,&■, Pig. 94, upon the picture plane. ^raw the hprizon line P<^ ; ► '" * ■ , I - .the trace of the given line it- { ^'.!'' oil the horizon plane is seen ,\,)' in y on the perspective: its horizontal projection isi X V ■ \f \s <■' Pig. 94 at th» intersection o- of , cil> by the line, Joining th^ foot of the station to the j ■r * foot .f of the perpendicular jf to XY. When the projecting plane revolves, c describes the arc of circle cc^ with ^ as- a center: the point 6f the given line corresponding to y moves in the horizon plane, therefore it will come in ^ o'^ the horizon line at the intersection wi«lfh the perpen- dicular 6;;^ to XY. The revolved line is J^cl, and " the revolved trace on the ground plane is Pig. 96 line seen in perspective in o(j^^ .s the foot of the station and XY the ground line. Join so/, produce to m/ and erect the perpendicular m^oc to ..XY: ol and Oy are the perspective and pro- jection of the same" point, A, of the given line* Draw the' horizon line HH'^: /is the pe^r- spective of the trace of the given line on the horizon plane. Revolve the projecting plane of the line around the vertical of «/ until parallel to the vertical plane: the point A of the given line, being on the vertical of ^^/^ does not move, and its perspective remains in ex. The per- spective J" of the trace on the horizon plane moves on the horizon line: when the projecting plane is parallel to the vertical plane, the perspective of the revolved line is parallel to the line itself and Tnay ire^drawn iirpe**^^ slnoa the ^ngte er*^yH — ±8 givent . The trace of the projecting plane on the ground plane TRACES 125 has come in ad,, parallel to the ground line. The point 4 of the horizontal projection corresponding - to d, of the revolved perspective is obtained by <, letting fall cT, perpendicular to XY and joining sQ. Revolving back the projecting plane to its orig- inal position, 4 comes in / and the corresponding point of the horizontal projection must be on the line 6-^ joining the foot of the station to the In- tersection /?/ with the ground line of a perpendicu- lar from S, But this corresponding point is the new position of d.,, and d^ moves on an arc of circle with ciy as center, therefore 4 comes in d and dccy is the horizontal projection of the given lirie. , The vertical trace Is found at c' by the usual construction: the vertical projection and horizontal trace may be determined as in § 16 or the triangle formed by cc' ^ cb and the given line may be re- volved around cc^ on the vertical plane. The axis cc' does not move, cl» falls on the ground line and the hypothenuse c'l*, becomes parallel to a^ . Revolving thrtriangle back to Its original position:;^ ^J, comes in 2> , which is the trace, on the ground plane,. tSiidtt, if^fi~- ~>^ a^ "/- * *^i_£ iiluLt. L i ^vl 1 J**"*^ ^ri i'>^* ^ . --i X 126 PERSPECTIVE «t of thfe given 3.1rte. Having now the twp traces, the vertical projection can be drawn by the usual oon- * - I struction. < 75. TO FIND THE TRACES .OF THE PLANE CONTAINING THREE ]$IVEN POINTS OR Two GIVEN LINES. Whether two lines or three points be given* the .prqblem consisting in ,',u: . AT' passing a plane through them is the same and oonarista.^ in finding the traces of the given lines or of those joining the given points* The -traces of same den(Mn- ination are Joined by straight lines which apb, the traces of the required plane* The traces of the lines are obtained by - 4-Y any of the processes of $ 60, Va or 74. In Pigo97,the heights of the. three points A,B and C are supposed to be known Pig. 97 and the traces are determined by revolving the pro- seating planes on the ground plane around the Jio«=. izontal projections ab and bo; ( § 73 ). QOit is f f ■ ! TRACES c the require^, plane 127 H / V. -^/^'^^ /> '' J\J Kg. 98 Sometimes the traces, of 'tte| plane are required » d . on the picture and principal plane. Revolve the principal plane around its trace, R^, Pig. 98,- on the picture plane, the front part of the prfincipal plane tum- Ing to the left. The station will ©ome In S. V •X V , Let Qr,y5,;>^ ' ^ be the perapectivea of three points, of which the projecti^ons on the ground plane are given, ^.^and o the traces on the pic:>ure .and principal planes of the horizontal projection ''• corresp^onding t.)cx,^; ^ and 2, those corresponding too:/'. Produce cxj to the intersection f with the principal line; /is the perspective of th6 trace y on the principal plane, of the line of space corres- ponding to the perapectivea^; therefore the trace^ on the reyoiyed j^j^i^n^ip^j plane ^Is on S/ But the .'^ \ ./ ^^ n-it. 128 PER3PECT1VB trace is on the vertical of c , therefore it is at ^', The trace of the other line is found in a aimi- f*»r manner at d^' and the trace' of tHte plane contain- ■ • / ing the two lines is c'el' - The traces of the two lines on the picture plane are obtained in a^' and b' as in 5 73 and joined to give the trace of their plane on the picture plane. The result is t^ie plane QRT. 76. GIVEN THE LINE OP GREATEST SLOPE, TO FIND THE. TRACES OP THE PLANE. The line of gMatest slope of a plane is perpendicular to the trace on the ground plane. Hence to draw the traces of the plane, find those of the line and through the ground plane trace Q/, r Pig. 99, draw «-Q perpendicu- lar to the horizontal pro- <^^ s jectio?!, cub, of the line: it is the ground plane trace Pig. 99 of the required plane- The trace of the plane on the =pictTire plane is obtaitted by joinings %^t^^ the ver tical trace, yi^, of the line- li ■ : 6 \ V ./*, ; V CHANGE OP GPOUNO PLANE ' 129 In Pig. 99, the line of- greatest slope is supposed .givenby its horizontal proje"^tion, ^^, ,and'"its per- spective kJ : The traces are found by' the method in 5 73. Should the line beknown by t^e heif^hts and per- .spectives'of -two of its points or' by tho heights and horizontal projections, or by its slope. the traces could be determined by the methods given in 5 60,73' and 74. 77. CHANGE OP GROUND PLANE. ^ A change in the ground, plane does not produce any change in the points or lines of the ground plan; the tracer of planes are displaced but remain parallel to the original trace. Fig. 100 shows the ground line moved^from XY to /f X ry; FigclOO X^ \ ;the left hand figure contains the projections of a point, of a line, and the traces of a plane before ths change -^of ground^TJlatiev— — In the first place, it may be observed that there t. ■'ix ^ fL-tj^^^^^u^.^i^ ^-« ^i^ to^ ci:} •tXi^)' Ci«*' 130 PERSPECTIVE lis no ohange in the vertical plane. beyond moving the ground line from XY to X Y^ . In the ground plane, the projections of the point, •■•. ' ' /n^, and of the line, ub , remain the same, but the Aft- trace of the line is now in c instead of b. The new trace is obtained by producinip; the vertical projec- tion, a^h' f across the old ground line, XY, to the new one» erecting the pery)endicular o' c and producing t^^ to its intersection with oo' . ' The trace of the plane has been moved from OT to 0^1^ .To find the new one. produce the vertical trace OR across the old ground line XY to the new . ,one , X, Y, ,and through the point of intersection, 0, , draw q T, parallel to OT. . 78. TO FIND THE HORIZONTAL PROJ-RCTION OP A FIGURE - FROM ITS PERSPECTIVE WHEN THE FIGURE IS CONTAINED IN A PLANK PERPENDICULAR TO THE PRINCIPAL PLANE. Take ' for vertical plane of projection the principal. plane and let QZ Pig.10l.be the trace of the plane contain- ft ing the figure. Take for ground plane the horizon- 1. tal plane passing through the point of Intersection of QZ with the tracd iJJljOf the picture pljine.XY will '. ''■.'tt&i''i., .■.t^rf.'fc t..- ■■.-:<>.-.i-,i.'.'- .^,AifAs£ahftJcii.:i-h&i4;Kmi^.kti-A^ :m^!mi&iM^M^rMi FIGURES IN INCLINED PLANES 131 z^, H .f, V ^K Fig. 101 be the ground line. Let S be the station, s the foot of tJie station, ti. n'f a point or the ('^iven fie;ure and /7i/,/74its perspective. The given plane, being per- pendicular to the principal plane, the vertical pro- jection of any point of the former ijs on the trace QZ. The picture plane, RQT, is perpendicular to both planes of projection, therefore the projections, of any point of the picture plane are on Its traces. Produce QZ to the intersection A with the ver- tical of the station and take SS, equal to ^A,S^ being above or below S according as A is bel^w or above ^. Join ^;nf/ and produce to the intersection h with the ground line: join Stz/ and nJf> . The Line S/t^ passes through ?w-', since m^ is the perspective of /«/ . The similar triangles rv'tn- q,^ rvsk give:- SA ' «S* / Jffom tHe "triangles h^nv, j*,S .we have: TS I 2 ■.V ■-it^i^^j(.vi|!^4_i,f>a5^i - .,. ■■ \t- ,.4j^ ,Mfcj,.. .; ' y^JU^iii^ksi jJi ^tr^ M r, ''».!' » w^^^«^w M ' lW ^ W» w^ l^ e^^^r'glg«^^Pl *^WW^!a^» Wt:^^ V ■ ■ * • «■' • " ■ ^ ' I 132 PERSPECTIVE But SA = jS, J> Thereforo: n^'Q. »Q. n'k l>s i ( m or, nOl ^^ thus a river at the bottom of a sloping valley should S^aHi£;-^J^W.S^>-i!v^-S^jS»i.i-jaL- ._;,.' .ag :■ '-i-.y,6jd^iBi^-!iJS.^j^Ktp^.-.3toJi!^^ ., 154 ' PERSPECTIVE be taken looking up the valley. • 79, TO FIND FROM ITS PERSPECTIVE THE HORIZONTAL PROJECTION OP A FIGURE IN A PLANE PERPENDICULAR TO THE PICTURE PLANE. The method of squares oT i 67 ^s-^, , Jean be applied to a figure in any inclined plane, by X conceiving vertical planes containing the sides of the sqiiares. The "intersections of these planes by the inclined plane form a series of parallelograms cor- responding to the squares of the ground plane. r^ Let QR, Fig. 103, be the trace on the picture plane of a plane perpendicular to it,XY the ground line, P the principal point, and cuhoel^ one of the squares of the ground plan. The projecting planes of ab and cd/ cut the trace QR • in ^^ apd 1^'• Through the station, S, draw a parallel to the inter- section of the pro- jecting pianos with the plane QR:the hor- izontal projection st * . > 1 , K tf ■k / -f^Y Fig. 103 '<> is parallel to cU> and .4^: »^ "^i< i^sM,^t^\ buM^ik^^^u^' re le > -.* >•>• ''{Ai^^HtJi . FIGURES IN INCLINED PLANES 135 I .- * cc^y ; the vertical projection passes through P and is parallel to Q.R, since all lines in the plane QR are projected vertically on QP. At t erect the perpen- dicular ^V to the f^round L^inerV is the vanishing point of the intersections or the projecting planes with the plane Q.R and the lines V//iy and V« are the perspectives of these intersect: ons. The distance *- 7?z/? can be carried on QR and as many parallels placed in-perspective as necessary. The same operation is repeated for /zyI^ and ^c^ and the figure cxj^y ^' obtained. on the per=«pec- tive corresponds to the square aJfoct. • / Another process consists in c^istructing the figure in the inclined plane by one of the methods of § 65.66, 67, usinf^ the plane of the figure as ground plane ( § 58 ). • - Let QR Pig. 104 be the trace of the plane of the figure on the picture plane, Hh' the horizon line and P the principal point.- " To construct the figure in the plane QR.that iJrJ^ng ^^^8 taken aa ground line: the n»w horizon-Jiine is J parallel H^ h/ to QJ? through the principal point. 12 f \- kiMil£y&^ . jM> H, «■' n 136 .„ PERSPECTIVE « t ■ The height of the station is the distance of these two lines.P/ii. The line which will appear as the .projec-A. ^ tion of the principal line on the constructed figure ' will be the perpendicular to the picture plane, at /^. On the real ground plane, the distance between the two projections of the principal line will be' equal to /;/,. X ; Pig. 104 Having obtained the figure us now take for groupd plane the of p, , the ground line being. XY. n the plane OR, let horizontal plane Q- wt R, _glg . 105 let ^ABCD, Fig. 105, be the \ figure in the plane QR, /n, l> the projection of the prin- cipal lihe.QR the trace of the picture plane and *, the foot of the station. TUe projection of A on the ground fil^ane 4^ at -thg aaae^- (li stance from the ground I BSsi^-r.- k FIGURES' IN INCLINED PLANES 137 !i line as A is from QR.but the distance oV the projec- «i . ■ tion.from s,/. is equal to wA multiplied by the ,»^ - /■ »-" ■ ■ co.sine of the inclination cc of the plane QJ{,for: Wt a^, Fig. 104, be> the vertical tjrojection of A and the right angle triangle p, a>rt gives: P^tV' p, ^ ' *~os. to ■ Therefore if A/w^, Pig. 105. be drawn parallel to Q.R, a^^ taken equal to A?w/ cos.«^ and the same operation re- peated for B, C. and D, the resulting figure a^o«^ is the ground plan of ABCD. The ground plan may be obtained in another way, for, join ^, A:the intersection cr with QJ? is the pro- , jection, on OR of the point of the perspective cor- • responding to A. TaJce ^, p' equal to s^^, sec. «. and ' through /^^ draw Q^ R parallel ta QR;join s,ay. The Simflar triangles ^, A «,,c„tA, give: ' . ^MB.i&m^* '^-^ ( 1 ) Prom the similar triangles J, /.'«,, ^,,, .a, we have: _TV4_--_4 y3 1 w. ,^ / ■ - J / V- -Mfc fc — — -- — — — ^^ — — — .- , Div^diTrg i ^~~f br r^l, peprafflng: 6;/^ and rrocu by 4^, sec.ct; and rnk qobco respectively , we find :' • , , • |& i rt(.»''l> » . ^i* * s , -A .4. ^ 1 , ^ -ii'-'jj. ^ ' v4^ „ '"■--•^ > ^ - 138 PERSPECTIVE This means that if the perspective oe moved in QR;«*, the directions obtained from the perspective for the different points of the plane ^ will be %%q direc- ^tions of the horizontal projections of th^se points. Therefore to construct the horizontal projection of the figure seen in perspective, find the distances ^of the various points of the figiire from the picture plane by means of a vanishing scale ( § 68 ) made with Pp, , Pig<,lQ4,as height of the s.tation and the real distance line. Then find the directions of the projections, using QR as ground line and a distance line increased in the proportion ^— r- • The figure constructed with the above distances and directions will be the horizontal projection of the figure in the plane QR. ■80. CHANGE OP GROUND PLANEj AND DISTANCE LINE. Let A,Pig.l06,be a point of a figure in a plane perpen- dicular to the picture plane and yj its perspective. Take the plane of the figuire as ground plane and let - ■ i t^e^ i ^ -J>e . the trace of ^the^ assumed i^infrlpal pljin e.^ V ^v- Revolve the principal plane 'around its trace on the .. £ t,i.\ ^^^^C^.^<^4''-^-B:r^K Vf^»^-t<»'X. ~'- i PIGii#S IN INCLINED PLANES 139 l^ground planertho station will come .in S, iJ and ^' are the projections of a and .<7-' the projection of A on the assumed principal plane-. Move the perspective / r: I i^-.^-^ Fig. 106 to ^ <^' 80 that s^p'm^LE— , ' u^ being the angle of the f assumed and real ground planes; it has been shown * . iihat ^^ is the direotioh of the projection ci' of A on the real ground plane supposed to be revolved around s,p, on the assumed ground plane. The visual ray, however, does not pierce the ground plane in ^, its projection on the princi- pal plane having been changed from 2>b' to S^v' by the displacement of thd perspective. But join ^p' and take as new ground plane the plane passing through , c : 5^7, will be the trace of the assumed principal plane on the na|r.ground glane and a!.; the projection of the trace of the visual ray on the last plane. Consequently the trace of the visual ray is at the -r /"•v inters ec t I^jtti from a4 to Jfo^' the pe rp §nd i cular drawn . " .■» At 'i- ^ J^' ./ 140 PERSPECTIVE Similar triangles f^ive the following proportions : /'' /It or and _yt^ (J a', P'n' = ^4/ P'u being equal to dc^, the figure p'cfu^r/ i^ a par- allelogram and aJ.a.' is perpendicular to s,a' , there- fore the visual ray will pierce the new ground plane in , 'ay. : i Hence, if the perspective be moved from p, to /y' a^d ,yV/; taken as ground plane, the perspective viewed f|om the station will correspond on the assumed around ^? p^-ane to the projection' of the figure on the rea] ground plane .-this projection can consequently be con- structed, by. the methods of §' 65,66,67. ' ■ \ FigolOe gives the proportion; I s c I - CcM- . w ^,P' °^ S^'= Ss, Cos.(v I The heights Ss\ S« , of the station above the r vai^ious ground planes being the same as the distance " -"Tor the Fi^cipaT pMnt from the corresponding ground Iii3^8,the tew ground and distance lines can be found i f^ ti 4.. .Jitsk.^ a-kt^tia ■65;^»i._ - m!*-^. ^v. .X. } i W FIGURES IN JNCLINED PLANES 141 as follows: ^ Let QR, Pig. 107 be the trace on the picture plane of the plane contalni4ig the fig- xire. Prom the principal point P, / // : '•// '•X'/' %/ // ^®* ^all ?p perpendicular to^ OR • and draw Pjt; and j^^ perpen- dicular and parallel to the real horizon line HH'o Taka Pd equal Pig. 107 to P/7 and draw Q^ R^ parallel to QR:it is the ground line to be used in the construction, because Pd = P^, a>s. w . At the distance point, erect DD, perpendicular' to HH'and draw PD, parallel to QR;PD,^ is the length to be used as distance lihe. » The height of the^statidn Pil^ used for the con- struct ion is always smaller than the real height P/?^* ^^y^bove the plane of the figure, therefore the precision of the construction ia less than if the figure had been in a horizontal plane- ' ' "'' ■ \ PLANE, TO CONSTRUCT THE HORIZONTAL PROJECTION tfP THE' ^■Hil, ..A .t ■^- :=e^ 142 PRRSPRCTIVB FIGURE" The mothod o( squares can be again employ- ed In this case Let QOR.Pip;- 108, be the traces of the plane of the figure o\\ the ground and picture planes, and oUtrrf one of- the squares of the ground plan. The projecting plane of fvcl intersects the traces of QOR in Q, and L:the vertical projection of the intersection of the two planes being La'. , Through the station draw a parallel to cui^ ^ It if .'the horizontal pro- jection is sy par- allel tp a^i/ the ver- tical projection is PV parallel to Lf ' and the vertical trace, V, is the vanishing point of the in- tersection of QOR with the projecting plane of ad^- The perspective of this inieraection is VL;the per- spectives of tho intersection of the projecting plane of c^ is VK and all the lines required may be drawn in perspective by eitrrying the distance LK on the Pig. 108 trace OR and joining the points of division to V. «' t. PifilJRES IN INCLIKED PLANKS 143 The perspectives of the interficctions with the plane QOR.of the proj^ctirif; planes of clO and od are obtained in a simj ier manner by drawing through the station a parallel to «/v, /j/R.for insta«aje, and joining the vanishing point v'to R arid T. The result- ing figure oLjhy^ corresponds, on the perspective , to the square abcrl of the ground plan^ It i9 also possible to construct a vanishing scale { § 68 ) so as to find the distances of the various poirits from the picture plane. Through the station, a plane \h drawn perpen- dicular to the vertical trace of the given plane: the Intersections of the latter with the picture plane and the station point are placed "in their actual po- sitions -and the vanishing scale is constructed by measuring the equal distances from the trace of the picture plane. 82. CHANGE OP STATION, (8>F!0UND ANT) PICTURE PLANE. The same result is arrived at by changes in the relative positions of the station.perspective and ground ane ^h % X ^i^E^ti^im^^m-^^^mw^'W^m^^^^^^^ plane, of the plane containing th(8 figure, whi ch we ^'*.^ Ji-f ,\ \ >A^ttT^« v~iV »,^*>--v- ^i^ i I ^ ' .«ilt)t3 ' " '•'it i I («5 pv^ ••t 1=*' f\f~l 144 PERSPECTIVE 'V' -4;,, ; -.r- K^ - ^L . . ^It i " ^ /? ^i^ .„_..^:1 --< '^ ^ '"-./' - //I /I Pig. 109 -?** J a will call A. Take for ground plane the horizontal plane passing through the \^^. intersectio>n p of this trace with the principal line and suppose the princi- pal plane revolved around its trace sj3 ■ on the ground plane. Let S be t^e station and a 11'* the perspective of the point mJrv, in the plane A*;, Take SS^ equal to <^s and suppos*e that S, be used as station in connection with a new plane passing through sp and the trace on the picture plane of the plane A. Call this plane B. The visual ray from the |iew station to ^,/a', will be projected Cut the planes A and B by a third parallel to the principal plane and passing through the point m^?^- The horizontal projection of bo^^l^^;w^/.^oijUti.^'.i.^>^^^ t'oL ■ - 1 ■•■ II -'-lS^lnPi-ii 'ti'^frvniN^n mh.^m '^=,'™P&",°*''A'-' \' ,y^^5«. \;i-, —- FIGURES IN IMGLIKED PLANES . 145 projected in n/cu parallel to sjv . - -^ , . Join S/^^ produce, to '//t-'', produce B^fx^ to its in- tersection with /z/a-/.>^>^'and n^a^^ to their intersec- tion v/itn S, Q. Join /fz^a^. The similar triangles give n^W ^ /nff*' (1) SD mfS S^E a.% (2) But SD = SE + ED - SE + yQ = SE + ^ S = S^ E herce ths f-irst terms of ( 1 ) and (2) are identical and we have: « Which is transformed -into: , > fi'S M^ The trlaagles SS, /x' and «>W/*' having one angle equal and the two sides about it proportional, are similar and ni'a. Is parallel to SS, . Consequently ay is on the perpendicular ;^^'/7«to s-^ ■ The line s??^, s^ « is the visual ray from the -^w^-statiim ttipowgfi: -the point T^T^^or-^He^^ tive: m.rv,a^^ ig a Uni of the plane B. These two iv%^c->:.44Jii'j^94it^ A ':^»^ » ..V / ^ L^ 1 )<. *h«K.^ /Si" 4< ^^® •. , PERSPEQTIVE lines intersect since the intersections fn. and cu of their projections are on the same perpendicular to the ground line and the point of Intersection is the ' trace of the visual ray on the plane B since the line^^/z., rW is in that plane. The same point is also the trac^ on the plane B of the vertical from rr.rr^J. Therefore if verticals be let fall from all the points Of the figure in plane A^their traces on plane B win form a new figure which will corr.espond to the perspective viewed from S . The problem isMThus reduced to construct from its perspective the horizontal projection of a fig- ure contained in a plane perpendicular to the picture plane, which is done by a change of ground and picture planes { § 78 ).The process Pig. 110 '^°]5U^"volve8 changes of station, ground plane, picture plane and trace of principal plane as follows :- Let QP,,Fig.ll0,be the principal litre . Revol ve the^ principal plane on the ■ i fi 'i 'i r i i i rt i n t iHu r .",""■. I -,—';s; -T" •-■I'TT"."";,-"' 'i " ""I'-' ' % w L ^ ^ '' KEFiteCTED IMAGES ' 147 picture plane around QJJ , the front part of the princl- pal plane being turned to the left: the station comes in S.and NS is the vertical of ' the station. Let « TQR be the plane containing the figure aeon in per- il spoctive. Draw Q* perpendicular to Q^^ and take SS, equal to -sTc Draw S, i^ parallel to sQ). The point /^>" F^ is to be used as principal point of the perspeo- tir"*^ ^^r ■^T)raw ?,p, perpendicular to QR, ///?, parallel to jQ and take ?, d equal to T^p. Through c^ draW ^ %K paranel to QR;it is the assumed ground line. Produce QR to NtQjg is the length to be assumed as distance line. On the constructed figure, the perpendicular to the picture plane at /?, will appear as trace of the principal plane" on the ground plane. i • ' The traces of the plane containing the figiire are found as in 5 76. _^ 83, REPLECTKD IMAGES.. The case of horizontal re- flecting surfaces is the only one that will be con- ■^ ■ When a perspective contains the direct and 1 1 ^ J 2 '■■ ir- 148 PERSPECTIVE reflected images of 'the" same point, the point can be located in space.plovidea the altitude of the station above the ref lee tirg surface' be known. Take for groum) ,piane the reflecting surface ani revolve tne principal plane on it.arounji its -trace. hQt^a^ ,czf^ Fig. Ill, he the point in space, tx,ci' its perspective and aa^ the perspecti ye of its reflect- ' 1 ed imase. The ^horizontal- projection is the same for both images, because, the reflecting surface be inp- hor- izcntal.the direct and reflected visual rays ar« i <-! ' ■ ♦ the same vertical plane having for trace Sk^!" ^ Let sar^^ S0<«/', be the reflected visual ray .'ac- cording to the laws of ref lection , the direction of SO is the same as if a/ were placed at a distance equal to o€iy below the reflecting surface and on the same vertical. Produce aJo to S^ : o^ being equal to ca^, sS - is equal to sB^. Hence, to find the position in space of «^, /7^;take s^ equal to j-Sljoin 3(x% s^; andj S^ 0;the point of intersection of Sor' and ^ is t\u ,ye rt J. c a 1 pj^eat4on. oj'^ tfee^ b o in^ ^f saace. ^^ le Join so( and produce to the intersectiort with ■M 'A: :;.,i• SHADOWS ' ' 149 - ■i Oy'cu, perpendicular to th«5 ground line; aaJ is the - ' ' • ' required point. ' » * ^ ' • The conatruotion gives 1 1 ' . " ' V. ,' '' not only the position cu, of. ■ ' ■ ' " " , „^ ,. ^j^Q point on the ground i.O > -^^ ■~^a plane, but also its, height T» _ CO.'. , ■ ' ' • The middle of the ver-r Pig. Ill .. . , .,• \ tical between the direct 1 1 g and reflected images corresponds to «<, the horizon- ^ tal projection of the poii;it on the ground p^iane. This > ' - ■ "•■ '■ i ' "' *''■ ' ' shows that when the shore of ft lake, for instance, is -1 •■■'->■.'. \ . ' indistinct on a perspective, it would be incorrect to / , * ' \ take for sho re line, the middle line befween objects and their images in the lake, because this .would giv& fof distance of the shore that of the objects them- selves. • ■ • .^ •• .: ^o--.:. } - ■ 84. SHADOWS. The subject of shadows is an important branah of perspective, but only those cast by tho sun need be considered here. „.,;.,i- At .1 it.. ■ ■ ■ me sun. xaxe vm equal to VB and from C and P Pig. 113- as centers with CE and 1 J Pli respectively as 2 _ - .'X. ■ * ♦ . J: '..,.... •' \\.. M,:.. X^::. ■ 152 ^PERSPECTIVE radii, describe arcs of circle. Join their point of intersection, Grto Cand GGF is the azlnjuth of the sun» When the perspective has been taken in the mom- ) ' • ing.plot the angle Z on the left of «/' in I'^'O.and the line Oj is the north and south line of the ground plan. In the afternoon, the angle Z should be plotted on the right of st^. The rules are reversed when the perspective of the sun is A^ove the horizon line. 85. HEIGHTS. In general, one perspective is not sufficient to detennine the height of a point, al- thoxigh there are exgeptions as for instance, points on the horizon lino which are at the same height as the station. The horizontal projection of the point being known, the height above the ground plane is measured divided into equal parts ( § 60 ). For instance at: and (J2^v ?ig. 114, being the per- ape ctivo and ho rizen^al projeoti on ef a point, and^r^ the foot of the station, draw a.V parallel to XY. ^ visa i- ..'w ' . ■_^y:zi:i-'^'-;^'z^'-yefMr-tr--- ■iixfA HEIGHTS 153 Prom the trace p of the principal line, take pB X equal to the diotance of a X ^ - ^' ''y f-" nt Pigoll4 from XY. Join ^B.and PE Is the height of the point above cty. . This height being a fourth proportional to three ^ known lines* can be found with an ordinary sector. Take with a pair of compasses the distance from oc to XY,pla^ one of the points on the division p of the jj^tor ( Pi,'^.115 ) which expresses the length of the distance line, and open the sector until the second point of the compasses coincides with the \ - \ ' corresponding division of the other branch, sp and SB being equal. Now take with the compasses the dis- \ ,- \ tance .from <2^ to Xt. { Pig.\ 114) and place one o^the points in p ( Pig. 115). The other paint being — t\. Pig. 115 placed on sp , will i \ \ Vaii i- ...•^'% 154 PERSPBCTIV^ coincide with a division of Jthe scale, B for instance; then turn round the compaasea and take the distanc.e from E to the same division of the scale jB:it is the height of the point above the ground plane. A • V ''m^ CHAPTER III / PERSPECTIVE INSTRUMENTS 1 i I 86. Many Inatrumenta have been devised for produc- Ing at once a perspective, either by mechanical or optical means. One of the simplest forms is probably the wire grating represented in ->>* ^ Pig. 116. Wires are is t retched on a frame so as to divide it in- to small squares. The frame is placed in Fig. 116 front of the object or view to be reproduced mi i*rt •V*. .156 PERSPECTIVE INSTPlJf.rENTS' and the draughtsman looks through an eye-hole in a fixea position. Dividing his paper into squares in the^ ^arao manner as the frame, ho is able- to reprociuco the outlines of* the subject by drawing; his lines through the- squares of the pap^r corr-spondinf^ to those of the frame. The distance from the frJame to the eye-hole is the distance line of the perspective when the squares of the paper are equal to those of the i>a^xe. 87. DIAGRAPH. ^Jhile for artistic purposes. the grating is quite sufficient, there is some uncertain- ty in drawing the figures of the corresponding - * squares. To obviate this defect, it has beeri imagined to follow the outlines of the subject with a pointer moved by the hand, as in Pig. 117. A drawing board, on nYiicY). ig stretched a piece of paper, is placed in an upright position in front of the subject of the perspective. It is provided with a straight ^d^e, supported at both ends by corda ^attached to a coimterpoise at the back of tli'3 board. . Tag si raigi-tt ^cr^ e may-4>e mox^e d^ Ttp -anti— doV-Ti or rightr- — ant] left, but owing W the mode o.f iM'^^'-micrL^ia "^■■; ^■,'"\f.-^?'n 'i-:! ■■" DIAORAPH , 157. always parallel t6 th« ?-/;,■: iiroctvon. The middle of * > "■ **ir^.n7 "^ the Btrai'ght edgS;" carries a r?e--ijil resting on the paper and thp.end has- a pointer rtar-rGsponding totho eye.-^'ole of the in;#trumoTit. The draught amaS takes the pencil with the ha«d an.d^^prcing his.oye at the *//' eyerhole.he follows with the pointer the outlines of the 8ubject,by.mo\ringthd^ straight eU^e in the fprop- er dli?ection. The petfcil repj^oducing; exa in front and those coming , from the. surface -Of^the paperi-JLt ^ »^iwr' y Pig. 118 is therefore X I • CAMERA LUCIDA 15^ possible tb follow with a pencil on the paper the outlines of the subject, the point of the pencilling seen directly and .the subject by a double reflection. The position of the eye must not iglry during the operation: to secure this, the upper face of the prism is covered by a metal plate with a small hole cut in the edge, through which the draughtsman has to look. With the form of instrument just described, the eye receives simultaneously impressions- from objects placed at different distances; the pencil is quite close while the subject of the perspective is gener- ally far away. The eye cannot accoinnodate itself to both distances at the same time; one of the images is always more or less confused and the work is very trying to the eyes. In the more refined instruments, the upper sur- face of the prism is ground in the form of a concave lens, giving to the reflected rays the same converg- ence as if emitted by an object tvielve inches awayc The paper being placed at the same distance below -.^hft-- pri B m»Ahe|mnoJa^« i d =^h »»ubJ»e^ -appear now-f tt-^^^ the same distance and can be seen simultaneously 160 PERSPECTIVE INSTRUMENTS ^ without any effort. The center of the pupil, a-, Fig. 118, is the station poir.t of the perspective drawn in P,and tJie height of Oy above the plane of P is the distance X line. But, the subjfecti, if looked at directly f rem «. would not appear as represented; it would be necessary . to move the eye to ^, virtual imafre of cu with refer- 1^ ence to the two reflectinf;; surfaces of the prism. A well constructed camera lucida is* provided with colored glasses, to equalize the brightness of the image s . - '-* 89. CAMERA OBSCURA. A camera obscura, in its sim- plest form, is a box hermetically closed to extrane- ous light, except that cordng through a lens placed on one of the sides. The opposite side of the box being in the focal plane of the lens, an image is formed on it of the distant objects situated in front of the lefts. Making abstraction of the errors introduced by lenses, the image of the camera obscura is a true per- .jLPj S ct i viLt j£jarL-,iA- xa-JthG - aaae ^ aa wouad^ « drawn oa « .. picture plane placed in front of the lens at a .\ , .. 161 CAMERA OBSCURA distance equal to the foical length. Let 0, Pig. 119, be the optical center of the lens art^ OA, OB, OC, three // .7 «? Ji rays of light com- ing from thifee dis- 1'^. tant points of space* j» ^0 Fifr.\^Q ^ The images of the points will form a triangle ABC on the focal plane R of the camera. The same rays of light will form, by their intersection with a plane Q parallel to R and at the game distance fr/)m 0, another triangle abc^, • m which every sicie is equg.'gi and parallel to th& cor- responding side of ABC^ therefore. ahc = ABC. The same demonstration applies to any other tri- angle and as a figure can alwtys be decomposed into ' a number of triangles, any figure obtained on the plane R will be the same- as on the plane Q, reversed. But the figure on Q ig the perspective seen from the _3tation__0 on the pic tur-e-jxia H ^^^the^ P e rt? ^hfriwage - \ of the camera obscura is the perspective seen from ,. \ % "%.' 1-^ 7 ^^ r 162 PERSPECTIVE INSTRUMENTS ■ „ ' the optical center of the lens on a picture plane ^,. * placed at ti^e first focus. The focal length of the lens is the distance lino. . . Before the invention pf photography the camera obscura was used for drawing perspectives and various /»'*- forms were devised t'o adapt it to that purpode. - One form consists of a prism W^BC, Fig. 120, with two spherical faces AB and BCand a plane re- floating face AC. The parallel /rays of light emitted by the '^ ' J subject are brought to a focus by the two spherical faces AB and BC,wlrfle they are reflected at right angles by the face AC. The prism is placed on the top of a tripod which supports a drawing board at the proper height to. re- ceive the image formed at the focus of the object * glass. The tripod is then covered with a. black cloth * ■ f ' ' ' to ihut ofr extraneous li^ht and enable the draughts- , ,. ,,-, ■ « . ^^tb*- 4i3Mig«^^ pro^ ected on ih€~ pap«P" *nd fol« ; =^ Pig. 120 low d^'i with a pencil >' >•> : . /vx _^'' / y ' PERSPECTOGRAPH v 163 The point, of the Ibncil being between the lens / ^and the paper, casts a shadow just at the ipoint where * the image is wanted. The instramont anown i%Fig.i2l is,^ot open to the same objection and requires 7 ' / neither tripod, drawing board nor even a black cloth- It is merely a box with a lens in fron? and a mirror ■.in PQ fnbllned ^ 45° to the axis' of the box. The .^■i -r- inaage formed by the lens is reflected by the mirror on a ground glass placed in AB. Being inverted a second time by the reflection, it appears • now upright. The lid which \ n \ ■ Li ■rd- Fig. 121 / . ' ^ .^%". covers the S^und glass when not in use. is open at an angle of about 45° with" the ground glass and cut^^tff sufficient light for the Image to bj vislbl*. in th|seJcoMtion3, the i'M not bright enough ^o work on paper and has tQ b^',- traced on .the ground glass, but with a black cloth covering the box and the head of the draughtsman.' it - , is Xff pTJWTbTir^l 6 worJ^ through thin paper. 90. PERSPECTOGRAPH. The perspectograph'^ls the . K 2 ^' |3? .4 ^SffS^ <: 164 tr V, Perspective tn-^Wuments invention, of Hermann, Hit ||r a\|erman Architect, fl^-?%.. ifk Its object being to '^rl* a Ferfpectivt ;^om m^ pi Of the siigect pnd not from the M)j^c|*&elf .^'Thfe ^j* lines of4lfl|plans ar^ followed --#1;^^' a tr§5p^M?%a • ; ' > perspecl part of ttaoi As^oorifii ncil -oig-rried . . V \ % I^ survey Ing, purposes, the instrument should be of small si^e.made entirely of metal;all th^ parts ,♦ , •■'#. should be' f fitted v/ith precision It should work easily and thejRpiount of dead motion should be as small as possible. !.,-.';• Although in its pre^ajt, form i't cannot ployed in connection wit^ tographic surve #- "^TrfTerspectograph, von Herraann Ritter,Archj%ekt' Frankfurt a.M. Druck von J.Maubach & Go. ' ,l^'' 'jw « \ . [i" \ V ■■,."*> ^ ^j. » ■ ^i^ M v.^ • ,.1^ ^i,^ "^^^ '}- -' ^ 4- / , V ■r ■!' t\ ,. '>. ■i: *^ ^^Jc !■ -,%■■•... „_jidBrj*.'3 ■m X I' • I, ^> *; ' K« r;t «^ '^i-yjM it! U- PBRSPECTOGRAPH 165 theory is given at lengthrwith some slight modifica- tions, it is applicable to any ol^her perspective ih- strumentt ' > However perfect sTn instrument may be, it always introduces in the final result some errors of its own Vue to dead motidh.to imperfections in the adjust- ments,ana to the slight errors unavoidable in the de- termination of the constantfe. Whenever the precision of the survey requires it, a geometrical cot^ruction should be employed, and the use of perspective |||fa^ru- ments should be restricted to' reconnaissance' or rough surveys, in which rapidity is more important than per- feet* accuracy. The principle of the apparatus is as follows: 'Let S, Pig-. 122, be the station, M a point of'^|^^«ound , ^ ' plane and a its perspec- tive. Take .^S, parallel to the ground line and equal to 6'S:join S^U. The similar triangles ' 4 .FiR-12» i.iSi&tii$il&i- 4-: .. isissMmsBssmm #■ A^^ 16b PERSPEOTI we have ProfQ the triangles s5^ M£ •MO NTS ' • '*, OWyM.also similar, ''iifei- '[p Hence, .%.. '^ But.by cgnstnaction, sS is equal tb sS, .therefore This property furnishes a new method for con- ^ structing a^ii^jBrapectl^p. Jake on the ground plan, Pig. 123, sS^ parallel to the ground line and equal to the height of the station. To* find *»thl4 perspective of^a .'•*•■• , « , ' in.'%|M of the plaii.join sU Is X- 1 *L^\4 A!i Y A' "iand S^M, which interseo#id line of the perspectlve.XY and take on it a point O' to represent the point of the ground plan. At erect Ow per?- \_j — pendirc^f^aR tor-X^ and e^iual "to -Ott^ j u t^ Xh^-ptrsTpwe^ f'JG «>.. i *> ' *^s ,« ^'^tive of M. Owing to the position of the figure, the w ' I:.. I in A. ' / w Is f" \ .''' PERSPECTOGRAPH ,' 187 perspeotlve [appears upside down. The perspective of another point, N, of the ground plan Is obtained in a similar manner, by taking OQ! e(iual to OQ and qV equal to Qk, . .This is done mP'?hanically by the peflBpectographi, .?M and S/)yi,Pig.l24,.are two wooden arms jointed in M and carrying the tracer. They slide through four ad- justable plecesiS, S, ', and «., : i^ and S, can be adapted •^ tp aiiy part of a rule RT, J' is fixed at the point of tiie ground plan representing the foot of the station ^d the rule or slide FT is firraty fixed to the draw- ^1% aboard, parallel to the ground line. The. second piece. S^ ,is placed at a distance of .^ equal to the height of the station and fixed in fi^t position. ' d which moves . in the groove of a slide N m> The third piece, 0, is fixed toj // fr'/ ^> \ ^ XY and carries a panto- graph system with axis at D fixed to the rod, so that the distance from to D is invari- Pig.124 able while the \ k 2 \ ■ , , ^v7^ 168 PERSPECTIVE INSTRUMENTS ] instmunent is in usd. When the arm 5-M is movedi ^ being a fixed point, follows the motion of the arm and carries with it along the groove XY the moveable rod ^nd the pantograph system. The fourth piece u, is connected with the joint A of the pantograph system, so that the distance ^u^ A is invariable during the operation; it is also bound to slide on the moveable rodk The pantograph system, is composed df four straight arms, AB, AC, Pi/ iand Pu', and two arms CDE and "BDG.bent at right angles in D. They are joined in A,B,C,D,E,F and G, Lhe sides of the parallelograms ABDC and DGPE being all equal. The arms Pu and Pu' are double the length of one side of the parallelograms and the pencil which is to describe the perspective may be placed either in u or u^ The sum of the' four angles at D is equal to four right angles: two of these angles, CDE and BDG, being right angles, the sum of the two remaining ones must be ecmal to two ri^ht angles, that is. — 1 ^ .CQB ' 4 . EneL-^-^a^aq--, — — . — But in a parallelogram the stmi of two adjacent angles ITn i ] r-. PERSPECTOGRAPH 1*69 la equal to two right angles, that is, GDB *■ DCA = 180° • hence, _; ' *f''^. ED(; ^ DCA. Therefore the two parallelograms are equiangular and their sides being equal, the parallelograms are equal, but not placed in the same direction. The diagonal DA of one is equal to the diagonal GE of the other," and BC is equal to DF. The line ^^' ' is parallel to,13E because Pu is equal to P^i' ; it is therefore perpendicular to XY since the diagonals of. a rhombus intersect at right angles, and it passes through D, because Bti is equal and parallel to GD. We have also »/' GE = DA. It is now easy to- ur.derstand the working of the instrument, The slide XY is placed on the ground line or rather on the line rcpres^enting the trace of the. picture plane on the ground plan. When the tracer in 4l-^8^moved on a parallel to XY, the arm M^ c ar rl es with it the moveable rod and the pantograph systttn \ I'^'O ' PERSPECTIVE INSTRUMENTS ' ,' ■ ^attached. The distance from to^,, doeg not vary, "^ since the sinilar triar.glee M^S, and MO^, "always give therjaM- proportton between O/i, W the constant length ^S,. The distance f roni ^, , to A and frpm to ''^^ * . n being invariable. and 0^, being constant. AD and con- sequently Dii do not change, and the pencil in u* de> scribeH a parallel to XYlit is the perspective of a ; line of the ground plane, parallel to thel^icture "" - planb. '. . ■ • - . . ■ .When the tracer is moved away from XY. in the direction of }As , th'«^ points ^d D do not. change, ■but 0^, is lenethened 'and^u, moves towards- the right carrying wUh it the joint A and .inc^asingHh^' di- ' ^onal DA to the s^me extent as^^Ou. ; Du befbg equal ' ^o DA.will also-b^Jlengtnened" and ^ moves down .pre- cisely th.'' same distance as^,. moved. liowards n'. --- The_ construction thus effected mechanically U ' the same as in P.ip. 123 . ^~The ^rAund line o- th^ per ape^tJ ve.xV; is the/p ine wh^ich Vould be. desoribec^by V tl?e pencil inu, if t^he t-rac^r Mc^opld be brou/rhf.tcJ: ' the .cent ej- of the grocwfe and moved on X¥: o-'and «>.,*- v would then coincide. ■ '' ' s . _ . . ... OV--'; V -+— »- '-K- >'^'.;%.^ ■■V:; "/-'■-'■■'■' ','; ' - < t- y PBRSPECTOGRAPH 1 171 h Drawing the tracer awa/ from XY . but in the di- rectipin su, ft^ separates from Csind 11 moves down /i '" by the same quantity, from its fonrtei*. position on V:, ■ the? i^r-ound li ne,p/i being pei^endicul.'-^r to xV and Qiji = 0^, - H i .-•■1 - >.-. » .'« ««> ■ * 4 S Now if M be placed on any other point of the Around . plan, the perpendicular Du. to xV'will be carried away the sa'ne distance as the point 0,and u will be at a '^^^^"^'^ frbm'XY' equal,, to tJ^o new value of 0^, . The perspective is upside down, the draughtsman Ijfevi^ ti3„.^lace himself i^ear U for guiding the trdSiar. '-^ T^e end -u*' of the arm Pm' describes the symme- j. . ' " ' ■ -' , J ; ' ' ■• ■ ■ ^' trical figure of the 'perspectfve, or the imager 'which would be seen in a mirror, but W6 re the* fixed point' *S ; placed on the left of .?, ^^, would describe the trui - perspective, the direction of the motidnfe of// and u '•■ ' '■■ -•" ■• -" .. % ■'^ ■ r ' . .being reversed. The ends ^ \ artd n' of Hhe Urtas of th.e ^ \y m pantograph system qre- both -fitted to receive the ^ pencil Which can. be changed from" one end to thtSither ^ "" ' >.,* as reqi^ifed. , '■■.,.. v ;'^ * » ^ -' • "'^ ,„ ^J\ ■ ; . i^ The instrument sdt as'- in Pig.lj?4 c,an Only -work ' ^. *0 172 PERSPECTIVE ■A ^#- possifile to place the slide xf on the other side of Rf,so as to work on points betv/een the picture plane and the station, but the obliquity of tho arms pre- vents them from sliding freely and the working of the instrument is unsatisfactory. ' It may happen thTt with a hi,n;h statipn and points at the extreme rirht or left of the station ( it would be the extreme left on the figure) the ■\ obliquity of the arm S^ M becomes too great to work. S, must then be changed from one side of .s" toi.thG other, ( from the right to the left of .9 on the figure) and the perspe-c'tive from one side of the ground line to the other. The pencil is at the same time changed to the opposite arm of the. pantograph system. ' The sliding rod XY may be reversed end for end in its groove, the pantograph system coming on the opposite side of the movable arm Ms. The pencil does riot require to be changed, but the arm bearing it, u for instance. instead of being between RT and XY,will -^"^bw be on the other side of XY. j<". ^■- >^ / A scale must be drawn on RTt the zero correspond- I K / PERSPEGTOGRAPH ^ 173 ing to the pointer carried by the sliding piece s ; the graduation extends to right and left and the pointer of the sliding piece S, is set opj^osite the division corresponding to the height of the "station -above the ground plane. The distance between RT and XY is equal to the distance line of the perspective. /, In the case of figures in planes which. are not perpendicular to the principal plane, it has been shown that the solution of problems involves changes ■ in the distance line: the e^es.of .^^b«,iirawing bpird must therefore carry graduations permitting to move XY by any giv6n quantity, keep jng it parallel to RT The different pieces of' the instrument are ad~ justable and must fiirst be place*' in proper position for the work in hand. This done, the slide RT is fi^- ly fixed to the drawing, board- ^nd XY placed parallel ^ to RT. There remains now to draw the scales and de- . . - termine the position of the v^rJ^ous alines and points ' .• on which rests the construction of the perspective Hitherto itvjhas -been assumed that the ' point's V, a ' - - % / \- 1 0. ^^ «t«T^ore m«tnematieai points and that thei: ^. ^% Cf ■^'s • ' A\^ r 174 PERSPECTIVE distances could be measured directly, but there is wothing on the instrument to define their exact po- sit ion and no such measures could be taken with p re- vision, corisequently,,these quantities or the constants ^ 1*^^^® instrument, have to be determined indirectly. Si. TO DRAW THE TRACE OP THE PRINCIPAL A^NE, ON THE DRAWING BOARD. It is assumed that the grooves in the sides RT and XY are parallel to their edges and that the sliding motion in these 'groove* is "also parallel, to the same, direction. This asstmrption is .f \ practically correct. Place S, above X /i X +- ■4" r .1/' /' r Sf U ^ (Pig«125) and draw a perpen- ' dicular 101, to RT, which, ff produced) would pass as nearly as po^fifibl^ through .•;". Should this line pass exactly through s, it would be the trace of the principal plane on the ground plane, and were the tracer U moved along the line from' M-to U , thepolnt or the slide, Xy' would' no.t move. To as- ,^. .■; '■'■ ■ . .• ■■ , '•■. . ■ „■ > certain whether has maved or ^o.t,marft .the position ^ Pig. 125 -. \ . DETERMINATION OP CONSTANT'S ' 175- of^ When the tracer ia in M, then place the tracer in M, andHf-^ ^^^^ moved to the rlghC, in u , this shows that Itt^.is too much to the left. Should MO and ,^M,0 be equal respectively to one half and twice the >i8ta«ce lih^,the error in the position of MM, would ; be equal to three times the displace«ent of^, but it is sufficient to estimate thd^uantlty 'by which MM. has to be displaced and to repeat the trial two 'v or thiree times. The potion of Ola indicated by the^ ^ ■ displacement of /» to the rigftt or to the leftia \. ,«0tl9n Of y, perpendicularly to XY indicates merely that S, is not precisely over f * - ■ h ^^2,-. Ta FIND THE DrSTANCE PROM THE STATION TO A FRONt' ^^m OF.T>^ .HOUNp PLANE. The trace of th. principal , . ^lane on the drawing-board being now determined, the , ground plan could be plac^. m i /proper ^osi ti cm ' ^ on the board. were it possil^le tc, measure- exactly, th^ di8tanawing board. Thi. ^ .^ detemination .is made a^ folipw^:- Drav. a seconj" .»« :Hne^CD parallel to AB;plaqe "th. two' lon^ anns one -above .t; ^tho r , ana ^^^> ^ ^ -'tracei^ tb >^^mark:.th< ^^ .i.^.Vv ..ft ' X. l"*'t; ^* %i ■|j*tV«.«M m^ ■.*♦!- v.. r- v-rriji'F^ . 1^^- 4 :u 176 ' PERSPECTI.VE position of ^. Then„ carry th^ tracer to CD and follow the line until . -^f' :,-, - . — ji^ — '/' the pencil has returned to the same point a ; ^ ^' " the line NM,if produced, ' "\ * . •■ ,' ,'i" would 'pass through 5 ^;, Repeat the same- oper- at ion in M^ N, and' ii,/ ,, Having the lengths of MM/ and NN, and the dis- tance a between AB^and CD, a simple proportion gives the distance a- from ^ to AB; . ' ■ * ' JC SB d/: ' Fig. 126 "■» The two lines and the. points NN^ must be taken as . >■■'■ far apart as the instrument will allow. ^ ' *^^^«..^ Having the distance of the front line AB.othe^r front lines at fixed distances from the *foot of the 8 cat ion are .permanently marked on the, drawing tooard.. The ground plan can now be placed bn the board "by putting the trace of the^prSnCipai plane in' 9©■in- eidence with the line dnawn |ffi the board And placing ■ ft ■i ■ t ,1 » I ■^.^ -(^.' ■ ,9 J: ^DETERMINATION OP CONSTANTS J77 ^ a front line of which the distance is known upon the oorresgondlng one of the drawing board. ' '93.' TO FIND THE DISTANCE BETWEEN THE TWO SLIDES. TI19 distance bbtween the two slides is equal to the '^^^^^''^^^^^e or the perspective: it must be indlpated ^y i fcale on iihe edge of the drawing board. In order to locate th^ zero, of the graduation the precise dis- tance has to 'be detertnined in one position of the in- strument.- ,, ,-; '. ,; ,/:; ' \ " ■'■'■ ' , V ■.. . ' '■ "■ '' "■': '' ^ ,, Draw a froi|r^ line AB; Pig. 127. Put the tracer ' n r *^ '0 (>, A y (icJ ft . firat in A and then in B, •*" ' aiai»king in each case'^fte . ^- "^ positions ^ and>^' occupied Pig-la?/-' ' ^ the pencil:'. Let d be the . ' tiis$«lrice between the slides and m the distance *f|N>fe\? to AB:we have: ^ ,^' ,ati,< /'«" AB s AB being a front line, Its perspective ^^ is par- allel to XY and equal to 00, , consequently; ire- s' •» .»j*» . »*, ilWi*iW» ri II I i m iii iy i / 178 PERSPi>£)TIVE \ , The three lengths which form, the secoV^d term, can be me|isu,red and the value of (fy cal-culated. ' • 94. TO DRAW TIIE GRADUATION "POR^Tfffi HEIGHT OP THE STATION. The heUght of the station is represented on the instrtcnent by jS, (Pig. 124). It is necessary to determine this distance for one position of s and S, , in order to draw the graduation for settihg^S, to ^any required height. Place the tracer M on the trace of the principal plane at a distance M-y equal to one and a half times sQ and note the place occupied by t^e pencil t«. In this position of the instrument we have » Then place the tracer M still on the trare of tge ■-"" X' A principal plane but at a distance M^- equal to three I times 50: we l/ave \ o^ = |-^s, / ^ The change in the value of Oa, i^d thus equal' to one third of the height of the station;btit thi& change is represented by the displacement of the pencil u., which Q&n be measured with a scale. Ttiree times thiii* ieme ht is the h eight »« LEi! a tatijarL . > 'i \ al es X' \ A > MgpRMI NATION OF CONSTANTS 179 The trace"r7instead of being placed at the dis- tariees from the foot of the station gi\^n aoove,may be, set at any~distance which may be convenient ; the fra-jtiun of tne nei>-^ht of the station obtained will ,bo different, but. the process is the same. 95. TO DRAW^Tlffi hORIZON,.GROLiND AND PRINCIPAL LINES ON THE JKRSPECriVE. The principal line is the per- spective of the traSs of the principal plane on the ground p|ane., The latter has been marked on the board ( § 89"^ ): following it with the tracer, tho^encil de- scribes the principtil line,. i ■ " J " % ■ • The jRroi^nd lino cir.not^'dravm directly, because *?; ■ . . I ^ t^e tracer] would havp td be carried along the front • slide and the consti-uctiun of ^the instrument does not permit it. The difficulty is oveAome by drawing the perspective of a front ii^e between the ground - " and ^^riz on line. '" - tet Pig.i:^ represent *L ^ 1... - ' p T^ J i — « L ^■: ■5M ; ^ ,- .." .,.,■,.,'•!' '■ .....,■■ .» . . , •' „ • ■ j ■. lane and U front line ane at a from the^jfoQt ■"-TiTf ii if ipi ^ i fnfrTni i rr w f'r^ 180 «; ■*■•*• u ** Perspective "'X-: "M' of the station- equal to twice the diatahce line". The similar t?*iangles S.vM, /*/'M give ' " /V^ 1 ^PI?"^P!^ n n f^ns 7^^^ '\^^ -■ >, „,.r \ **' -* '"I ' ' ' t ■■ .. _>■■ " ) ' if - *. Ss is equal to the/height P/' of t^e principal point above the groiind line, therefore, » .. ^ •„.,'■ *' "" ■ , ' " •■ Following then with the tracer the front line drawn on the board at a distance from the station equal to twice t^^a^^^^ce line, the pencil will de- scribe a horizor^l^HMSpe midway between the, groxand '^ fiuid horizon lines. One half the height' of the station is now measured on each side of the line so obtaij and parallels drawn to it. The line nearest to the - ; >',.. .. front slide is the ground line, the other one is the. • horizon line. - ' . Ss Processes similar to those given foAthe per- spectograph can be employed for the other perspective instruments. With the camera lucida* however, particular at- tention must be paid to the fact that the station is . not the same for the direct and refleoted^ rays § (88). rl I %,r \ ^\.( r ; '^ CENTROLINBAIJ 181 Although the fact ha^no importance when drawing the .. PtrapectiVe of remote objects, it irou«[ jtf .not allowed for,eau«ie cqnsldarable. errors when the 'subject and the .perspective are both caj| to the^ptation. ' ^ 90^ - CENmLINEAD. In addition to the instrumenta . >lready de8cril>ed, others have been devised merely' to facilitate-; the (}o.n»tlE*uction of porapefftives. They ^ , are nof properly apeakinfe, perspective instruments, / ^ine^ they do nol^, erfable the draughtsman t6 draw the J- perspective directly. ' -; ;. the vanishing point df a^ line nearly parallel to the"|,icture plane being; at a great distance from the ' rlSrlncipalpoiht:«ay fall' outside Of the paper, in which casejspecial constructions ^re necessary to draw a lln.; which.if produced, iould pass through 'ihe point. -L The^'centroline^* permits to draw a line vanish- irig fit any polnt.no matter how far f rom„ the principal point. It consists of a straight edg^. (Fig.i29 ) with X^p arm^ whose>clination to the straight edge can be varied at pl^«^,re. TWo studs sL or eight - inches apart. are fixedto the edge of the drawing *l board. The y " '•■% -> 1%'j ama .QX^thfl eontro^i > H>y^is^lttg^y^^^^ / II gg 1 . Bgjg / '■■■■, 1 ■ " ■ ^ ■ .V, ■ ' ' J g .^ • • .1 --, • ' 1 K. ' '' - - -" \ t 1 ^ 1 ■ / • • " \ ^ '- - • - . / ^,. , • * !■- » \ \ ■ ■ ft - - . 1.^-' .. ■ B » c 1 ^^^'l ■ •*■ ' ' 'Hi HHHH| |M| ■■■■ HHIjl ^^^^^^H^^ 1 ^^^^^H ^^^^^1 ^^^^^1 ^^^^^^^^^^^H^^ - '^^^^^^^^t ■ ■ ■ ■ P ■V ■ ■ 1 ■ F^ / ■■/',:"' ' -; ' ■[ ' ■■ . ,./ — ,.,... ,| X ' ' / , - " ;'■' ■ " ,■ . ♦ X "^ * ^^^^^^^^jr / 1 ; f-- ^ -* - - . > , J, c '^.- ' ^ ■■ * \ - '' *•' " " ' " ■* ■ ' ^^ «" ■ ' .• ' . . .: , rf J ' ' \i ^ 1 ' ' -M. 1 ;U- 'i^^ ?V- ■ ■ /■' " « ^ fe *' -I'.v ? , •• - 1. . >dA 1. ''■'"'■' 1 ■ » . ■' ■ .-■ N ' , /-^ _,^^l. ^ • riin'iiiii'ii iiMJiiMi ^^^ ..;......,^.„.... ^^^ ejIL • IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 23. 12.5 m m^ 11-25 iu 2.2 2£ m 1.6 Sdaices Corporation '^ ^\ .' V ..- ... lii^Ti/.', t • * hj BK=3 182 PERSPECTIVE contact with the studs the various directions of the (•»• Pig. 129 ' Straight edge intersect at a common point. Let OC, Pig. 130, be the straight edge.OA and OB the arras, A and B the studs. Through A,0 and B.pass a circle: the arms being fixed in a certain position, .. the angle is constant // » V-r- - -r-\/.0 ■y-' --^' Fig. 130 ■r // and is bound to move on the circxunference of the circle whenever the po- sition of the centrolin- ead is changed, as from OC to oV. -^gi^?^oduce OC and cfc' to i % mi^mm/W' ■ ,*. i CENTROLINEAD 183 their second intersection with the circumference: they must cut it at the same point V, because .the angle AOV being invariable must always subtend the same arc AV, no mattei* on what point of the circumference the apex may. be. Gonsfeqnentiy the straight Wge will draw all the lines vanishing at V. * The centjJ'olinead is employed only for horizontal lines, whose vanishing poiht is on the horizon line* The stud's A and B are placed on t'he same perpendicu- lar to the horizon line and at equal distances from it; it follows that the horizon line HH is a diameter of the circle, and, . . VA = VB ■ ■ -i-i The arms are equally inclined to the straight «die. The line OC* bisecting the angle AOB must pass through V which is the middle of the subtended arc BVA. The distance of the vanishing point, V, can be 1 varied eitjjei*''by changing the positions of the studs or the inclination of the arms. Increasing the dis- tance AB between the studs, the size of the circle is ) increased in the same proportion and V moved to the 1 loft. i 12 V , - t } \ 184 PERSPECTIVE f It is not usual to disturb the studs, the changes in the distance of thie vanishing point being obtained by altering the inclination of the arms of the cen- trolinead. Were the arms perpendicular to the straight edge, the vanishing point would be at in- finitum and the instrument would describe parallels ^^^o-Hhe horizon line. Closing the arras gradually, V comes nearer to AB; when AOB becomes a right angle, the intersection of AB and HH' being the centre of the circle the distance of V from AB la one half of the latter. Closing again the arm8,V continues to move to- wards AB without ever reaching it. In reality, the studs are not mathematical points, but cylinders: the direction of the straight edge is, however, the same as if the anna rested against the axes of the cylinders. The direction of the vanishing point may be given by a line of the ground plan or by a line of the perspective. In either case, the arms pf the cen- trolinead have to be set to correspond to the vanlsh- ing point. '♦^^ \% ♦^^F I CENTROLINEAB 185 In the first casei revolve the picture plane on the horizon plane around the horizon line as an axis. The station coraeS in S.Pig. 'Y! Pig. 131 iSl.SP is the distance line. A and B the two studs. Let SV be the direction, of the given line on the grotmd plan.V is its van- ishing point. Through A.B and V.pass a circle; the centrolinead should be set so that the straight edge, being on DH, the arms should be ' •• " • on DA and DB. - . Join VB;the anglfe VDB, inclination of the arm on the straight edge, is equal' to the angle VBA, because they subtend equal arcs. Join CS and BS,ahd draw Uc ■ and ct" parallel to AB and HH';Join„^f . By reason of the similarity of triangles, 3k is parallel to VB \' ► : and the angle j^^e = VBC 1 f • Therefore place the straight edge on MB, the axis of rotation on fi , and adjust the left arm of the cen- trolinead to coincide with ^a' . The other arm may. /,.,., . .^y- . ■ 1 !^Saes^**!«!tettKS?=-«'v»-.- 186 ^ PERSPECTIVE - be set by placing the straight edpe on f^^t the axis on ^ and adjusting the arm to coincide with *M,or better by placing the straight edge on the horizon line, the arm already adjusted in contact with the stud B and moving the other arm until it comes in contact with the stud A. The lines BS.CS.Mc and ck^ are drawn once for all, ^y need not bo drawn, sp that the only line to be marked is Sy, direction of the giveri line on the ground plan. TH/hen the given line belangs to the perspective 'f"- the const met ion must be / / slightly modified. VE.Pig. 132, being the given line, take any point, P, on the horizon line, join "PE and FB, and draw pU parallel to AB. Through e , draw ^k parallel to EV and join k^ . On account of the similarity of triangles, /^ is parallel to VB and the angle f^&c is equal to VBA, inclination oiSthe arm on the straight edge of the centrolinead. // Fig. 132 V PERSPECTOMETIR ■ ■■L 1.87 V FB and o)K are drawn once for all but PE and /^ have to be marked for every given line: that is, two lines instead of one by the former construction. Centrolineads are usually sold in pairsione to work on the right of the principal point and one for the left. 97. PERSPECTOMETER . In § 65, a method has been given for transferring a figure from the perspective to the ground plane by means of squares formed of lines par- allel and perpendicular to the ground line. The "per- spectoraeter" has for object to dispense with the con- struction of the squares' perspective. On a piece of transparent material, glass, horn or celluloid, draw two ■. V . ■'*■ I D J) g52fi± A 9 O n ni p m' 7i'~^nff Jl Pig. 133 parallel lines AB.DD", ( Pig. 133" ) and a common perpendicular l?p . Take DP.PD', pk and /CB equal to the ' distance line and from p, lay on AB ^1 equal distances •JUii 188 PERSPECTIVB prtv, noiv, prtv Join to P the points of division' "l^n, ttC, n!',. ... part of those lines intersect AD and- BD at« ■ A ■ • ^'^J O'^/ ^*^® corresponding points are joined to- gether by lines which are parallel ^o AB and DD< !rhe instrument is now placed on a perspective, with P on the principal point and DD' on the horizon line. The ground line will fall in XY.for instance: 'it is divided into equal parts by the lines- converg- ing in P and the figure of the perspectometer is the perspective of a network of squares in the ground plane, having the equal parts of XY as sides. By re- ferring to § 65, it will be seen that the construction is precisely the same in both cases. This instrument is useful for restoring from the perspective a figure of the ground plan. By. placing it on the perspective the, squares covering the figure are at once apparent and only those required are drawn on the ground plan. The side of the squares is equal to the length intercepted on the ground line between two of the converging lines: this distance is laid on the ground plan from the trace of the principal plane smd par- jf^ , PERSPECTOMETER , 139 allele drawn. to the trace throygh the points of di- vision. ' v The front line nearest to the ground line is laid on the ground plan either by estimatjing its dis- tance from the ground line or by dofns true ting it. The estimation is- made by-noting the fraction of a square's side which represents the distance from thp ground line. , . ,. "" Figures in planes inclined to the horizon but perpendicular to the picture plane s»re transferred to the ground plan by placing the centre of the per-, spectometer on the principal point and , the pa4'allel . lines -in the direction of the trace of the inc|[feed plane on the pic tur* plane. The trapezoids of the ',.%^ instrument are the ferspectives of squares in the inclined plane, which squares are projected as rect- angles on the ground plane ( § 77 ). The longest sides of the rectangles are perpendicular to the picture plane and equal to the length intercepted between two converging lines of- the instrument on the trace of the inclined plane. The shortest sides are the projections on the ground line of these inter- '} { •. • 190 PERSPECTIVB • oepted lengths. % The rectangles are constructed on the ground plan and the. transfer made from the perspective as in ' the preceding case. ' ' » 1 When the plane containing the figure is inclined to the picture and ground planes, the principal point must first be -displaced on tHe'p'rlife,ipal line so as • -■V \ . to project the figure on a plane perpendicular to the picture plane and having the same trace as the given 1 plane, ( § 80 ):the problem is now the same as the last one. ' • The perspectottieter can only serve for perspec- tives having the same distance line, such as photo- * graphs of distant objects taken with the same lens; - every distance line requires a new instrument. The ' wi/ith Fp should be equal to the height of the horir- zon line above the foot of the picture; the length DD need not be larger than the picture, the distance points being placed on the figure merely for the ' purpose of demonstration. The length of the equal parts of AB should be such that the divisions of the lowest ground line *( • .. " ■ ■ ■ ; • ■ ', - -, Li iLiii- DRAWING THE GROUND PLAN * loi employed be not too large'^for the degree of accuracy required. These divisions are the sides, of- the squares or rectangles of the ground plan ,arfd the larger their size, the less accurata''''wil]Lthe transfer of the figure be. The instrument can be m^e by drawing it on large scale on paper, and^aking a reduced negative from which a positiv/ is ra*ae on a transparency plate. The transparency xd bleached in bichloride o'f mercury and varnished: th^ lihes originally black, are now white on clear glass ' 98. DRAWING THE Jl^^OUND PLAN WITH THE CAMERA LUCIDA. The distinction between the' picture and ground planes is purely conventionalize picture plane may be taken for ground plane and the' ground plane for picture plane. If oi be the perspective of the figure A of the ground plane, Inversely A is th4 perspective of the figure (X of the picture plane. Gotisequentjy any perspective instVument can be employed to draw the grotind plan from the perspective, by a change in the setting of the instrument, the distajice line being ■ '■ ' ' now what was formerly the height of the station and 4^ i .mmxff. 1&2 PERSPECTIVE . \ \^'\ ■ inversely the new height of the station being the foAier distance line. - • , «"' 'With the camera lueida, the prism must be fixed ., permanently so that the height of the virtual ste^tion SP,Plg/lo4f above. the plane on which the perspective is placedTbe equal to the distance linp of t-he per- spective ( 5 86 ). As long as the latter does not cliahge,the prism must remain in the same posi- tion. The per- spective Is placed in HH'xy, -i!n nuoh a manner that the line ' SP joining the virtual station •s K Pig. 134 to the principal point of the perspective be perpen- dicular to the plane of the table or drawing board. ♦ _ ■» , The ground plan is on a p]!litform sX^ Y^ , parallel ■*^ to the plane of the tableland which otfn^ie moved up and down.- It must be so placed that the perpendicular c^ • •s \ ^ ■ .1^ ' ' DRAWING THE GROIWD PLAN - 19» 3y from the raal ptation to the piutform, bfe equal to "tho hcj-gM of the ^station; s in the foot of the ■ - 3tatlol{»jind the ground line .13 somewhere in X, .Y, .' ' The eye looks by reflection at the perspective, and sees directly the pencil on the platform. - . ' The determination ^f the -constants -is made by methods similar to those of the pers^ctograph ( 90, 91,02,93 ). The caniera luclda permits -^ :i^-. "•^ #. . :1^' ■■ 194 PERSPECTIVE INSTRUMENTS ■ s . r r yy y r Ji '• { I J- Pig. 135 the height of C5f '( Pig.124 ) must be equal .to the height of the station, and JS»^to the distahce line. The principal point of the. perspective^ rsiust be placed in s, the horizon line under RT and the ground line under XY, The Instrument, thus arranged would not work with the perspectives used in surveying, it could not even be « . 0- DRAWING THE GROUND PLAN 195 set.thn obliquity of the arras being too great. It may, however, be employed to transfer a figure of the per apccti.ve to other planes than the f^round plane, as for instance to obtain the elevation of a building from the perspective of the facade; the method is fully described in Fitter's pamphlet ( 1 ). In other cases, and particiSJi'ly for topogi^aph- ical purposes, the pencil must be placed in Kl and the tracer in it. The perspective is placed under u, with -the ground and principal lines on the lines previous- ly marked on the drawing board ( § 92 )• Taking M wit^ the hand, the arras are moved so as to follow the lines of the perspective with the tracer it. The operation being precisely the same as for drawing the perspec- tive from the ground plan ( 5 88 ) it is evident that the pencil in 11 will now reproduce the ground plan. The use of the instrument in this manner is at first a little difficult, owing to the point M being guided l3y the hand while the perspective has to be traced with^, whose motion is entirely different; some .<& I Ar ( 1 ) See note, page 158. M 2 .<'j»Mmmmt-j*i- . 196 PERSPECTIVE INSTRUMENTS practice is wanted before being able to handle it succeapfully. A certain amount of dead motion is inevitable in* // Y • y, .?t t X *'/ ^/ \i^ \f> i"-rA. S V Y' ^^ f X' // /' // i.V/ Figol36 an instruraomt of this kind, particularly when changing the direction of the motion perpen- dicular to the slides XY and RT. In order to avoid the errors which this' would introduce. the pencil M should always be moved in the same •direction, away from XY for instance. Wl^ion the draughtsman comes to a part of a line or curve which is directed towards XY,he should lift the pencil, push M back to the other end of^he curve and trace it in the opposite direction. The position of the horizon line HH'of the peij- 3pective,Pig.li56, varies evei*y time the distance from 5 to S, is changed, for it corresponds to the tracer l*. when M is at infinittmi and' the two arms parallel: 11^ would then be in N. Now change the height of the t ffffv DRAWING THE GROUND PLAN 197 station from S^ to ^, ;N comes in Q, carrying with it the joint A of the pantograph system to which it is rigidly fixed, DA is increased by NQ and jn moves do-^n the same quantity. So the horizon line is displaced towards the front of the drawing board the same dis- tance as the station -is moved up. The ground line is not affected. The instrument is provided with means of adjustment for the distance from A to u .these two points being connected by an iron rod sliding in a ring at A, to which it can be fixed by a clamping screw. A graduation placed along side the rod permits to add to All the increas^, in the height of the station, in which.case, the horizon line of the per- spective does not move and both the perspective and the ground plan occupy invariable positions on the drawing board, no matter what ground plf^e may be used. When the distance line is changed, as for figures in inclined planes, the simplest manner to place the perspective on the board is to put the pencil in M on the trace of the principal plane ( Fig. 136. ),M0 being equal to 0? ; then slide |(^ perspective under the tracer until the latter is on a point midway between m ^'''*!JifflfMilii^iMfaiBiflrfcd'j^Atvifc ■i^^.mA^i^.^ 198 PERSPECTIVE INSTRUMENTS the ground and horizon lines. The principal line of the perspective rm^st of course, coincide with the line previovT^:lv drawn on the drawing board. ' 101. CHANGE OP SCALE. The perspectograph set as in Pig. 124, wi 11 only work on points beyond the picture; plane. It is possible, by placing the slide XY on the other side of, ^T, to reach within the picture plane, but the instrument does not work well. It is preferable to use it as set in Pig. 124 and to resort to a change of scale' when figures within the picture plane are to be operated upon. . Let ^-XY and XYX^ X ^> «"'' Pig. 137, be the ground and picture planes. Take a nenf ground plane, s^X^ Y^ , ^ at a distance ss,^ below Pig. 137 "the other plane equal to thp height of the station S.s' , the figures obtained on the new plane from the perspective, will be on a scale -double of^ the former scale, ( § 53 ). The new ground line X, Y^ corresponds to the front line K$> of the plane .s-XY, midway between V % DRAWING THE GROUND PLAN 199 the foot of the station, 6V and the picture plane, 30 that it will now be possible to worK with the per- spectograph on the part of the ground plane comprised between AB and XY.but the result has "'to be reduced o to half size. By doubling again the scale, one half of the space between AB and s, will be covered; the result must be reduced four times. In practice the draughtsman would commence oy working on the figures beyond the picture plane. After transferring them to the ground plan, he would move S, , Pig. 124 so as to double s% and draw a new ground line on the perspective, by doubling the distance of the first one from the principal point. He would then place the perspective in proper position for the new ground line and continue to operate the instrument' as before. . - , The restored ground plan is drawn on cross section paper, having squares of four sizes, distin- guished by lines of different intensities. The largest squares are divi(ied into four smaller ones and the latter are also divided into four. The sides ^ m 2 ?7* --!i^Ab^^»^^?«k.:L-u'^ i»<.v 'v»t»^-^...t£^ t^ AA.a.^itt «r *l fi t* . * -'. A. a ..4*. Jk»^ -.t&kS r^ ■iH -T^^^i' . ^ j.^a^n--.- -^ >3^if.fi ■T *r> a. . ' / "s 'I \- ' -V 1^* . '» *^-* ^ I1P« ^ ■! li ALTAZIMpPM 203 circle and telescope ^la given by an endless screw, which can<^e LifteLd out of bearing for rap-Id motions. The telescope may be revolved erld for end in itg Y'm ; the verniers of the vertical circle not being attach ed to the oirclcthepp are two pairs of them, ono for each side. There are two levels, one for the horizon- itriding leve^^ tal circle and a 3 >'The usual coloured and reading glasses are provided. The defect ,of this instrument ^hich is common IT to nearly all those of the same class, Qonsiats in its ntumerous adjustments. The>^ are particw^.ai'ly ob J actionable for surveys in a mountainous country where occasional rough usage ia unavoidable. It is much to be d^irpd that the make r^ flhould supply small instruments w«ithout any adjus tments; they could be made sufficiently accurate for ordinary work,aitd for precise operations, a surveyor does not trust to his adjustments Ih any case, but determines the errors of the instrument and applies the corresponfling cor- rect Ions to the observations. 103. ADJUSTltENTS. This altazimuth being of an un- usual pattern, it may not be out of place to describe 1 ■''■'. ^ ,:'iTBi,J.iM , ^^ X 204 KIELI) IN3THi;MP,N'i'3 ItB adjustment M . t ^ The vemlera ot i. vertical olrole can be moved up and down by neariB of acrewa:th«y aiuat firat be placed in the middle of their courae and the teles- cope raised or lowered by its supporting screws un- til the vortical circle reads at the same time 0<* and 180°. The stvne supporting screws will displace the telesQope to the ,right,.^r left and care must be taken that the verniers impinge equally on the grad- uation. 'The lovel of the. horizontal circle is adjusted as in othe|* instruments. The next point is to ascertain whether the two collars of the tel^acppe have the same diameter and, if not, to .determine the difference. 'With the slilding level, level the upper face •• (JD of the telescope (Pig. :7.j^gPlioul# the two col- f not b^^^qual, the lower face AB will be inclined to A f r A" A •H « ■ Pig. 138 the horizon and the inclina- ALTAIIJ^TH 2(55 % ,t loir of \he toloacope'a axis wUl be one i-'lJ' that of AB. '^. .^^0 W^^ % Rnyftrso the telescope In ita Y'a:thc u^;^;.'P fsico J,»!B|&%J%. pdi<1 tlon Cb', or which the i noli nit ion !•? £ ^ >, ^>f ■ , _ > <^*(iyf® that of AB. 'The atrldinR lovol being set on gD WiX.1 be out by a certain number of divipior.fl which is the measure of the Incl/lnatl un of CD'. One half that quantity la the angle dV the two facea ol th6 telescope or more precisely » the anple ol' the (^one dfj- termined b^ tho two collars. Then level the vortical f:xla of the inatrument by the ordinary raetliods, and witl; tho aupporting screws of the. telescopeiinove the latter until the atifiding level indicates for tno upper face CD, an inclination equal -to the angle 6f the two facealthe "l^er face AB is then horizontal. The inclination of the optical axis is one half that of tho upper face; it will be the -same when the telescope is reversed, but in the opposite direction. In adjusting; the sup- porting acrews.it is necessary to pay particular at- tention to the verniers ancj to aee that they overlap i 'A equally .on the graduation of the ^\rr/l^. >> ■*9y ft. ,t'^ / /*.' ,v '^,:^- *:i . 11 ^,d.' I- i.*^ .^!2._ .Mj&^mMm^r ^6 ^^ • FIELD INSTRUMENTS > , ' ^ The wires are now placed ih focus for distant Objects and the draw tube of the telescope fixed per- manently by the screw proviiecr for the purpose. The priam is then taken off , the telescope set on a distant object and turne.r round in its bearings. ' The wires ravist be adjusted until their inter- section remains constantly on the. same point while tfie telescope offectc a complete revolution. The op- tical axis then coincides with the axis of figure of the telQscojJe and the point sighted upon la on the »•■*' aaffiie- level as the instruraent, if the collars of the i, ' ''*' telelcope have the same diameter; if not, the altitude. of the point is equal to tiio inclination of the op- ^ tical axis, which has bepn measured. Now replace the prism in front of the object glass, set the telescope on a distant point and re- verse it in its bearings. The vertical wire will pro- bably not cover again the same pointlaorrect one half of the difference by the adjusting screws of t^p prism and repeat the operation until the point is ^ biaocted by the vurticwi wire-totn^eforelmdlitiBW reversal. The line of slgh^t is then perpendicular *N »x ALTAZIMUTH 207 to the axis of the telescope. The next adjustment is that of the verniers. Set the telescope on the point used in adjusting the wires and which is in the horizon of the instrument. The readings of the verniers should be 0° and 18 O^ : move them until they read accordingly and turn the Ihstruraent ISQo in azimuth. Setting again the teles- cope 'on the same point, the readings should be 180° and 0« if the circle were exactly centered on the axis of fl^re of the telescope. This being seldom the case, the line 0° ISOo which before reversal coin- cides with the line AB,Fig. 139, joining the O's of the verniers, will be in aV after reversal. The vernier A will reaji f6r instance 179° 50' and B.Qo 10'. The verniers must be moved over half the difference: in the present case they should be made t9 read 170o 55' and O** 05'. After turning the In- O ¥' ■JXp- ISO /I It Fig. 139 -irirument ±8tr-m aziimith, they will show the same figures. 80 that the altitude- of a point will be given i!£iikMiM.j, i^mtm^2m>e*^' 208 FIELD INSTRUMENTS at once by taking the mean of the two vernier readings in one position of the telescope. T}ie point sighted upon was taken in the horizon % of the instrioxnentrany other point may be employed, its correct altitude being determined by the mean of the two altiy^gifes obtained by observing in one position of the instn;!ment^ turning it 180** in azimuth and ob- serving a/^ain. Instead of setting the verniers to 0° and 180° in the first instance, they a,re set to the correct altitude and the adjustment is completed in the same manner as before. 104, TFUPOD. The tripod is a sliding one: it draws to a length of 38 inches which experience has shown to be sufficient in a mountainous country: it weighs 3>^ lbs and when packed is 20>& inches long. It was ad- opted at i'ir.3t in the absence of anything better, the intention beinp: to replace it as soon as a proper one could be procured. Contrary to expectations, it has proved so steady that it was decided to keep it. The tripod servos for the altazimuth and the camera, both ifes-truments boing secured by the same ■ screw with spiral spring, which forms part of the ••iSf^ ALTAZIMUTH 209 U tripod haad* Three small levelling screws passing through the heeul permit to level the camera. Both instruments being very light, steadiness is secured by a net between the tripod legs, on which a heavy stone is placed. Vith this device, not only are the observations and photographs better, but there is «l&o risk of the Instrument being blown away during :t * one of those wind blasts so frequent in the mount- ains . 105. USE OP ALTAZIMUTH. The instrument being set up and levelled, the surveyor sits opposite one of the verniers of the horizontal circle, his assistant sitting at the other vernier. Turning the prism of the telescope to the right, he commences by observing the points immediately behind him. For doing so he bends to the left and looks through the telescope whose axis is directed towards the right. Without leaving his seat he reads the two verniers of the vertical circle, the vernier of the horizontal circle and enters the readings in his book. The assistant ^eads^ia v& miar mad notes it in his own book^- The same operation is repeated for eveiry point im Ki N '^ ,^'i^fuiM» f---WW- 2^^ FIELD INSTRUMENTS on the right of the surveyor. until those iiaaediately in front are reached. For observing them, the sur- veyor bends tp the right and looks towards his left. The prism of the telescope is now turned towards the left and all the points on that side observed in the same manner as those on the right* The surveyor thus completes a series of readings around the whole horizon, ISOo being with prism to right and ISQo with prism to left. The instrument Is then turned 180» in azimuth. the eye piece of the telescope being on the side of the assistant, who observes to the right and left as did the sm^veyor before and enters in his book the readings of the vertical circle and of his vernier of the horizontal circle. The surveyor merely takes the reading of the vernier of the hor- izontal circle in front of him. Two complete series of readings around the hor- • izon have now been obtained. one with prism to right and one with prism to left and during the whole time, neither the surveyor nor the assistant has left his Beat . Shoul djpore^preoise obse rvations be wan ted, --^^-^ the aame operations may be repeated after reversing ^^»6\ ?pppp ^ is t ■WsWi^-j J » , 3 ii m- 'Y > &i ^\M H Vc -i "S. 211 0AMBRA3 the teleaoope In its Y'a. After oon^leting the observations, the mirveyor and his assistant compare notes: every point having been observed and every reading taken, independently by surveyor and assistant, any error is discovered at once and rectified on the spot. 106- CAMERAS. A large number of cameras adapted ^o surveying, have been con8trueted;although some of ^t^em are very ingenious, they have not come into iseneral use. They are fully described by Lieut. Reed < L. r .^ jil "Photography applied to Surveying". The instrument employed by the General Staff of ^e Italian Amy (Plate IV ) is an altaaimuth and ^ O ^amara combined, the object being to measure with the • ^telescope and graduated ciroles not only the angles required in ordinary surveying, but also the direction of the prinqipal point of the perspective. The vert- ical circle and telescope are on the side of the in- strument, and the camera which is of metal and has the fom of a pyramid, is in the center a^ove the horlsontai «trclirr IT can be^e^ched frdnTthe al- ~ taalmuth when not in use. N 2 -< i- r ^ M ir 'J O J XI c o ; 812 FIELD INSTRUMBNTS The German pattern, Fig.140, ia due. to Dr.Meyden- bauer:ir is also a metal camera, in the fonn of a pyra- mid, set on a graduated circle wj^th levelling screws.' The views embrace an angle a little over efi^ and the mode Qlf using it consists in turning the camera pre- cisely 60O in aziffli^th, after each view, by means of .the graduation. so as to obtain a complete panorama with six views. It is. Intended to be employed alone, all angles requii«fed for plotting the plan being taken Tram the photographs. ^ . This instrument was . devised and employed prin- cipally for making plans of buildings, for which it may be well adapted. For topo- graphical purposes, a sep- arate instrument for measur- ing angles is a neeessity, so Dr.Meydenbauer's eanwra eould not be used alone, as - Z^-^y _ =.^AnteBdad^ With tha twfr ia- ^ » t # s^nauints, tha graduated CAMERA 213 t # circle of the eamera becomes useless and only adds to the bulk and weight. Great superiority is claimed for the metal camera over a wooden one, on account of its greater precision* Without entering into a discuss- ion on the ooHiparative merits of wood and metal for instruments of pi^cision, it may be stated that a camera of seasoned mahogany, well bound with brass, will for; topographical purposes, give all the accuracy that photography applied in a practical way, is able to give. The possibility of taking a complete panorama with six vietrs does not prevent any important advant- age. On the Canadian Sxirvey a. seldom more than two or three views, together embracing an angle of 120® to 18.0®, are taken from one station, or are found to be useful. r A" siurveylng eamera merely requires to be simple, strong, and without any adjustments liable to get out of order, except the means of setting the sensitized plate vertloal, which Is a necessary condition to ob- tain correct p»rspeetlv«a.^Any eaa»ra fnif llllnf^ these requirements is adapted to surveying. r;.--. 1- .\i 814 ritLD INSTSmCUITS 107. IiBHSSS. The oholeo of the lena is of great iJBportanee;Bo eatisfaetory reaulte can be expected with an inperfect instrument. ^ There are several kinds of lenses eniployed for •5 / landsoaile photography ;tfome oonsist of a single aehro- oatie lens and others of two eooOjlnat ions, either •iailar or not. The single or landscape lens does not give cor- rect perspectives: if, for instance, parallel and per- pendictaar lines be ruled on a sheet of paper so aj^ to divide it into squares and a photograph taken with the paper parallel to the sensitised plate. the i«Age does not consist of squares, but is a figure either \ • \\ j spindle or barrel-shaped* aooording as the diaphragia is behind or in front of the lens.' In photographing / a subieet with such a lens, .1, a .dh^ the only vertical lines rep.- 1 n ■ resented by straight lines, are those in the principal plane; the perspeetives of other vertical 9 ar^ ottrves > 1 Mi. . Pig. 141 • aore and more inclined as « • \ Itwia CAmRA 215 the dfvtanee from the principal point inoreaeee. A^ essential ooi}dition of a eorreot perspective is that all yertieal lines be represented in the perspeotive by parallels to the principal line, therefore the . photographs taken with an ordinary lalidacape lens are notr suited to surveying. Fig. 141 represents an ordinary landscape lens with rotating diaphragm. Mr.Dallaeyer has brought out lately a landscape lens which he calls "reetilinear". It is probable. ' '' t 'v hpwevertthat the distortion is merely reduced without having disappeared entirely. The landscape lens has over all others the great advantage of a smaller / \ number of reflecting surfaces and minimum thiekness of glass: less light being absorbed by the glass and lost by refleotion»the Images are so much brighter. It would be the best lens for siirveying purposes»lf it eould be suide absolutely rectilinear but that does not seem possible. The lens most generally employed is a combina- tiOT^ of tw o aimilar lenses batweea whleh the di a ^ phraipn is placed. It has reoelved different names n 2 •^iiMDmm,mm>^n 21d FIELD INSTRUICSNTS from the naJcerai sueh as the "Rapid Reotllinaar" of Oallmayer* *Rapide Rectllineaire" or "Apleoie'tlque* of Francals and Hermagis* ■Aplanat* of Steinheil» "Buryscope* etc. i but all these lenses are construe t- ^ ed on the same plant althoujc^h there are differences in the curves and kinds of glass employed. To understand how the distortion is corrected. ' the double combination may be supposed to consist of tvo singla landscape' lenses* between which the dia- phrapn is placed. The front lens. having the dla- phragm behind oauses a spindle-shaped distortion which is counterbalanced by the barrel- shaped dis- tortion due to the back lens having its dlsphra^n in front. This must be taken only as a rough explanation of the len8;a doubla eombination does noi consist of two landscape lenses. and the eorreetlon of the dis- tortion is obtained by ealoulating the curvature to be given to the lenses in order to produce the effect de.«ix*ed. In the •rapid* variety of this form, the distance- between the two combinations la about equal to thelo* diameter, the diaphragm being placed in the middle. ..^ OAMBRA / 217 ?ig. 142 '^xy-^' This arrangement permits the use. of a large aperture. the lens bMng what is eaU- • cd a -"rapid* one. The de- finition, good at the centre, decreases rapidly towards the margin of the plate and can only be improved by the use of small diaphragms. Any point near the centre of the plate recelTes *. '- - the light coming through the whole of ihe lens* open- ingjbut owing to the grelit distance betfeen the two combinations* part of the light is intercepted by the mounting when the point is near the edge of tite plate. The restilt is a rapid decrease in the brightness of the image from the centre to the, edge* which can be improved only by the use of smaller Istops. Pig. 142 represents Henry Hermagis* "Rapide Rectilineaire* lens with attached diaphragms. In other lenses constructed on the same plan* the distance between the two combinations is reduced mdre orJLesB:they are designated jmder the general r name of "Wide Angle" lenses. The diaphragms b^ng o i::3te 4;fc:*a^!fe#?.»*-«#?i»-. 218 ^nSLD INSTRUMENTS close to .the lenses, their diameter has to be reduced, so this form is slower than the rapid rectilinear. On the ^her hand the diminution of the distance between the two ccxnbinations .causes a oiore uniform distribution of the light over the plate, the bright- ness of the image not chang- ing so rapidly as in the • rapid rectilinear. The def- inition keeps good at a ' greater distance from the centre: if a rapid rectilin- ear and a wide angle be stopped down to the same aperture, it will be found that the latter will give good definition over a * larger area than the former; it will cover a larger plate, hence the name of wide angle. Pig. 143 represents Dallmeyer*s wide angle lens with rotating dlaphra^n. Comparing the defects and advantages of the various lenses. the wide angle is indicated as the Xana Xor snrv s ying pttrposys* Rapidit y is mrneeesswy— ^ Pig. 143 but uniform lllumiimtion and definition, in which the u t 1 ?«'»WPW|»:-' '■,'J> "lii JKi'', u CAMERA 219 wide angle excels, are iiaportant. A well construeted wide angle lens is practically free from distortion. A simple way of testing it consists in dividing a sheet of paper into squares by parallel and perpen- dicular lines. Place it parallel to the ground glass of the camera and take a photograph: the result will show at once whether tfeiSre is distortion. With a good lens no error should be seen without a micros- cope: the measurements and constructions on the photo- graphs which will be described later on being all made with the naked eye, it follows that the distor- tions due to such a lens may be neglected. The terms of "back focus* and "equivalent focus" are often found, in manufacturers* catalogues and in books relating to photography. The "back focus* is the distance from the flange to the focal plane: it has no particular meaning beyond expressing the ( length to which a camera must extend to be used with a certain lens. The "equivalent focus" is the focal length of^ 1 ' twr i Ihgle lens which would give the same image as the lens referred to. It Is seldom given accurately ..Ar!n'.„MISl^:J.. . ■■iffit^itfta& .^ua rt. i fc . 220 PIBLD INSTRUMENTS in the oatalog^e8• To find itidraw a geometrical fig- ure such as a square or triangle* on a plane parallel to the ground glass. focus and move the camera until the image seen on the glass is equal to the figure itself:mark the position on the bed of the camera* Then focus on very distant objects and mark the new position; the distance between the two marks on the camera bed is the "equivalent focal length* of the lens* In elementary text books on optical matters* several processes are given for measuring the focal length* which rest on the assumption of an optical centre through which all rays of light pass without being refracted* For instemce it is suggested to J) measure the distance -between a figure and its image wheh both are equeil and to divide by four: this method depends on the hypothesis of the optical centre. It must not be forgotten that there is no such point, that it is merely a convenient hypothesis for faoilit- ating demonstrations or explanations and that any Tnetnoffdi^BnoiiigTipoTrirrgrveB Willys a^^ result. In every lens or combination of lenses* there ^ CAMERA 221 are twb points called "nodiil points* such that any incident ray of light passing through one of the nodal points will emerge from the lens In a direction parallel to the incident ray and passing through the other nodal point. The equivalent focal length is the distance between the second nodal point and the focal plane* The brightness of the image varies proportional- ly to the size of the lens or to the square of Its diameter. The larger the aperture^the more light is adndtted* It varies also inversely as the sqtmre of the focal length: thus if the focal length be doubled, the aperture rAnaining the same .an equal quantity of light is admitted but is distributed over an area four times larger, the brightness of the image being reduced in the same proportion. Representing the aperture by w and the focus by y*, the brightness of the image is therefore propor- tional to -~ r This fraction is the maaauro of the rapidity of a lens or of It^ capacity to produce a certain lu> "-inii^mummt^t 222 PIBLD INSTRUMENTS minoua effect such as the Inipression of a photographic plate. The aperture referred to is the effective aper- ture of thejlens which may be nnach smaller than the diameter of .the lens. It is customary to express the aperture as a fraction -^ of the focal length: then £* is the rap- idity and the exposure of a sensitized plate must be proportional to x . The use of stops or diaphragms is equivalent to a diminution of the aperture and consequently »l)f the rapidity. It must be, supposed that the effective aper- ture is equal to the opejjing in the stop: it depends entirely on the position of the stop with reference to the lens. The following process due to Steinheiltpermits to measure the effective aperture. Focus on a distant object and replace the ground glass by a screen with a hole '/s inch in diameter»on the optical axis. Put a light close to and behind the hole; covering the lens with a piece of ground glass, an illuminated circle wl3,l be seen which represents the effeetlv«^op«filTig. - — - 108. CAMERA 09 CANADIAN SURVEYS. The camera enploy- CAMERA 223 ed on the Canadian Surveys (Plate V) is merely a rect- angular Tdox of mahogany firmly bound in brass, one face having a hole fbr the lena and the opposite one being left open to receive the plate holder, llae size is the English half plate, 4% x 6^ inches. The box is con- structed with great care, the faces being perfectly plane and as nearly parallel and perpendicular to each other as they.csui, be made. It is ffxed to the tripod by the screw in the tripod- head rnuts in which that screw fits-, are encased in two of the' faces of the camera, which may thus be placed with longest dimension either horizontal or up- right. The opposite faces are arranged to receive two levels at right angles:they reat on the' face Itself and can be changed from one to the other according to the face which is uppermost at the time. A diaphragm in the middle of the box, cuts off the part of the light admitted through the lens which does not contribute to the formation of the image. Four fin e &ombgT on e aighth of «>inehwid»»«pa ==- fixed to the camera imnadlately in front of the 'f.'^.- r ih 224 ,> TIBLD INSTRUMENTS ■ * « • middle^ of the slides of the plate :they serve to fix the position of the horizon and principal lines. The combs being fixed to the camera and not inside. the a holder, are about one quartier of an inch in front of the plate: the distance being short and small stops used in the lens, the images of the combs come out sharp on the plate although not in contact wit]ti,it. A hood with sun shade can be adapted to the lens, for cutting off the light not required for the image and protecting it from the direct rays of the sun. The lens is a Dallmeyer wide angle.No.l.A^ »of 5/< inch focuslwith the half plate employed, it gives an angle of 459 in one direction and 60° in the other. Three levelling screws, forming part of the head of the tripod, serve to level the camera. Once level- led, it may be turned arotind, the screw fixing it to the tripod acting as an axis^it remains tolerably level during the revolution. Two lines -Qoarited on each of the faces receiving the levels, indicate the angle embraced by the instru- r yoR A q gge w h at he Is ta king , without the use ^of the ground glass and black cloth. \ » * \ 'S'S^-'J "Pl^^*; ^ -^V^WiA '■" i .*S'-'-'»^J?yn*-SrW»-o'S!j- (tr'rs<*»^»ji^ ^ ■\ ^1 1 */ I 1 -' ■: >» * Ml ^•1 o.,^£^u.n4-&.;--&^'^a-g^J^^44'g'^^f'^^s^,;,t,.^ iiriwJVi iii»i.mfciii tyj - '( II ^ s 1 ■ 1. '^, ■■■ Aci^Miqpanyl ng the camera are six double, holders • oonta;j^nir^ one dozen of plates. The fapes of the holder's are nxambered from 1 to 12. 109. USB 07 CAMERA. Having set up^ the camera on the tripod and levelled* the surveyor turns it around lantil the lines on the upper faoe show him that it is properly directed. He then looks a^long the lines of the other face to see whether the viaw will reach high or low enough. If not, he puts the longest di- mension of the camera upright, if not already in that position. He now inserts the plate holder and draws A the slide, after maJcing sure that the cap of the lens irB on. He levels again and takes off the cap. The exposure completed, he puts on the cap introduces the slide and withdraws the holder. The views are always taken with smallest stop they are pexiiaps not so brilliant as with a larger stop, but the details are finer and that is of more inqjfortance for sux>veying ptflrposes than artistic appearemee. l Ti U--^ ADJUSTMBHlS^AHD S^TEittaNATXQR 0? GOKSTANTS OF— THB CAMKRA. 'There is but one adjustment for the 226 FIELD INSTRUMENTS camera: the sensitized plate must be vertical. The levels being on the upper face, the box inuat have been made with sufficient precision to ensure the vertic- ality of the plate when the camera's upper face is horizontal. The verticality can be verified with a plumb line or with a level mounted like a builder's level. The following process is more precise. Put a piece' of plate glass or a good mirror in the holder, in the place occupied by the sensitized plate, and open the back. slide. Set up the altazimuth near the back of the camera, fix the telescope to an altitud* of 0° and by turning it round, find a point on the same level as the telescope and which can be seen by reflexion in the plate glass of the camera. Then turn the instrument round the vertical axis until the reflected point is seen. If at the intersection of the wires, the plate is vertical. For let A Fig. 144 be the projection of the in- f strument on the principal plane, M the projection of jthe^p^int. and.-BC 4fee^ t raaa-oT the r e flec t^ing surfaoq= of the glas8«plate;A and M being on the same level. :.t-'-^^l^^^gmss^ CAMERA r 227 AM is horizontal. When the telescope is turtied towards ' the glass plate, the horizon- tal line AD is the prqjec- h ^ "al M A Pig. 144 tion of the telescope's op- tical axis: should CB be ver- tical or in other words should the plate be verticali a ray x)f light emitted by the point M and falling in D woxild be reflected in the direction DA of the op- . / tioal axis. Should the top of the plate be inclined back- warda>as in figureiMD will be inflected in DP and the point M will cease to be seen on the horizontal thread of the telescope. In order to bring it thereiit will be neceasliry to raise the t^escope so as to look in . ». . the direction AG.IM will ahoir above the horizon. Should A '' I ■ ■ 1 the plate incline in the opposite direction the tel- e^ . The distance line is equal to » f sin/. CO and the principal point is at a distance from /> equal to P COS. CO The horizon and principal lines pass through certain points of the comb, marks which are in the middle of the sides of the photograph, having been im- presasd there by the combs fixed to the camera, a short distance in front of the plate. These points are noted and will serve in future for drawing the two lines without any new determination. ^ik t^ ».Ii J IS,-. 1* k^ ■p CHAPTER V PHOTOGRAPHIC OPERATIONS 111. DRYjPLATES. The views employed in photograph- ic surveying arp those of distant objects: they are , the most difficult to obtain by photography. The fore- ground is easily obtained clear, bright and full of detail, while the distance will come out fogged and in- distinct, unless' the best plates be employed and skill exercised in the exposure and development. The dif- ficulty is due to the blue haze interposed between the lens and the distant points: blue being a very actinia. colo rTa #l.ight haae un per oept ibj.e t o^ the ey»r" has a most damaging effect on tho negative. ^ ' : I ..i.,"«'k»!.„..». , :A:\\ ran-ii i i i - i mi »8»!WW !.t"IM I >i»'»'W> .I l l i > I 'i I -I ' •7 DRY PLATES [• > 255 . . ...\ It is impoaaible' to giv^ preoia|e .^directions con- cerning the kind of plate to r? usedi^photogiraphy is progressing so rapidly that m a short time thf? di- rections would become ob?olt.te. I A collodion qrthoehromatic emulsion has, it is / saidf. been discovered recently; if reliable, it would be , the best fo'r surveying purj.>osea. The blue mist would not' so much affect an orthochromatic emulsion as an ordinary one and the collodion ougiit to yield fine negatives for enlarging. , ."1 Of the plates now ^ on the marke^.the laott auit- * ^ ' '■■ • * J" ' ■ ' * able are probably the gelatino-brcnii^ide oi*ihoohromatic dry plates. '^The ordinary gelLatino-brpmi^e plate is most sensitive to blue and violet rays* and little • affected by green, yellow '.and red.. "A special' treatment of the emulsion or' plate makes it more unif orally , w ■ . • • sensitive to all colors; the plate is theil sajid to be •orthochromatic* but the full :effe«t of the treatment requires the use of a yellow or orange transparent screen somewhere in front of the plate. By the pro- — s a aa te Ari ^ i^h I t i ta a bewi a j ufa mi i feed -tti e pl aty has- — —^ become more sensitive to gree^iyellow and reA rays; ;f '.. /< .t v^ 236 PHQTOORAPHId OPERATIONS the colored .screen, in cutting off .a large proportion of the blue and violet rays, increases the effect and .^. finally, the different colors are rendered in nearly their true values. a^ !»• Among the ordinary plates, the slow ones will -^ give the best resnits. They should be thickly and . evenly coated and allow a very great ^lafiitude of ex- ;^ poaure. Except the borders covered by the holder, which should be perfectly clear, very litUe clear glass should appear" in the finished negativera plate in which patches of clear glass are seen yields hard negatives without detail in the shadows. The grada- tion of the half-tones should be continuous, from nearly opaque black to nearly clear glass. The plates ; mast keep well, and be easy to develop without accident. These qualities are found in several of the slow plates now on the mdrketrwith a long exposure and a careful, well restrained development, they will be found to work fairly well.- During the last few years, a great number of films of various itinds have made their appearance. In a ;;v' ■■r.^'- r-i ?;■ ;";■.:.;■■"".;':■'. "" 1 . ' > * mountainous country, nothing better could be desired; M^ DRY PLATES \ 287 large sizes could be used without increasing the , weight to be carried and without the risk of breakage. Unfortunately, there are aerioua objections. In the first plaoe, no filio has yet been produced which may be pronounced equal to a good glass plate and several require after 'treatments which oaake the process long, tedious and liable to accidents. Then they are more or le«a subject to contraction or expansionias long as it Is uniform, it does hot affect the accuracy of the views wnd their fitness for surveying, but it is not certain that thla unlf^ormity exists and the least distortion is sufficient to cause their rejection. The packages of plates received from the man- ufacturer, are tranaparted in tin boxes hemetioally ' closed: the boxes are always kept locked. A Chinese lantern made of ruby fabric, is used for changing the plates. This is done at night in the surveyor's tant. Should the moon throw too mueh light, a few blaxAets thrown on the the tent, make it safe enough. The old plates are takan out and replaced in ^ttw original paek«ees;tBo iHnri>Tateff~iire~H»?*ea:, isw troducad in the holders and duated with a wide camel's » — i'^r^-i I:' r 838 PHOTOGRAPHlV OPERATIONS hair brush. The holder, are kept eerupulonaly free from duet ln.lde;other,i,e the constant motion would l"lh« It on the plate and cause plrtioles. Before Introducing a plate In the holder, the n-ber. of the do.en and of the holder ar^ln.orlbed m pencil in one of the comers. >i^„. .rltten -Uh a soft pencil on the fil^ ere <,uit. pl.i„ .,ter aevelopment and have no chen^cal effect." 112. nPOSUHK. The lengtk Of exposure 1. mnuenosd by four causes: 1- THe rapidity of the lens. 2. The rapidity of the plate. 3. The strength of the light. 4- The nature of the subject. f The first two causes are constant and their ef- fect may be detennined once for all. The rapidity of the lens is the aquare of the to tha^^^ength. The lene and stop used being al- ways the sa«e.the rapidity is constant. Ihen ordered, it Should be apecifi^ that ^^ Plates are to be of one ennUalon: their rapidity win «*! ♦ u ■,. '■■."'C "■■■■ '■?''"'^'^"?,^^^' XXPOSURB 230 f-i ♦ be uniform at first and will change )but little after- ' . ''' trarda. Having ascertained by experiinent the time ol ex- posure required for a certain subject and with a certain light* it may be assiuned that th'e same ex- posure will suit any similar subject with a light of equal intensity • The strength of the light ia measured by instru- ments called photometers. One of the simplest con- sists of a piece of sensitized paper on the surface of a tinted oapdboard , the tint being that acquired by the sena^^tlzed paper after some oxposure to light. The photometer being exposed to diffused light, the sensitized paper will acquire the same tint as the cardboard after a certain number of seconds: this number indicates the strength of the ligtit and it is assuQied that the exposure of a dry plate should be proportional to it. This is not strictly true', the sensitive surfaces not being of the same kind but fl is sufficiently accurate in practice. * ^DeeoQdnn*^ photometer consists of a screen of graduated opacity interposed between the eye and thuj^ -^p »<■ fi*0 PHOTOGRAPHIC OPERATIOMS ground gla.. of the camora. By inoi^asing the opacity, a timaeoneB when no light is .eenrtbe opacity then indicate, the brightness of the Inrnge. The graat ad- vantage of thl. photometer is that It measure, at the •*»• time the Influence of the last two eau.e. of un- certainty in the time of exposure, the .trength of the light and the nature of the subject. It is open to two objections. In the first place the eye 1. in- flocnced by the light itself a,ul what appear. opaqu KNLARUING \ 243 film during the various opchatibna to vnhlch the plate was submitted. The number of tHe plate already in- acribed in pencil is now written in irJi in the margin 80 as to print with the subject. The plates are kept and stored in Qnvelotoj|^|rf>Troperly numbered apd class- ified, feftf^fe' / 114. ENLARGING. T^Kw^t can be made either contact printing or by enlarging. Contact' printing undoubtedly gives the finest prints, while details are sometimes lost in the process of enlargement. On the other hand, measurements are taken direct on an enlargement with a precision which would require the use of a microscope on the contact print; the enlarf^e- raejit also affords room for the coij^Piction lines which would soon become confusing on the smaller print. In so far as the perspective is concerned, an enlargement is the figure which woujld be obtained on a plane parallel to the picture plane but at a greater distance from/the station: thus if a per- a p a e 4ive be enlarged-few lgfrrirt^wirH:"t?t)rrg3pond-~t-o~^ picture plane with a distance line double of the \ P 2 .•/1.W \t , \ ij»ff miueeaiii.i V 244 PHOTOGRAPHIC OPERATIONS ' real one. ■ In the same manner a photograph enlarged twice is the same as one taken direct with a lens having a focal length double that of the lens employed. It is convenient to enlarge in a proportion which will make the distance line an even length: for instance- with the negatives taken in the camera described above, the enlargement makes the distance line one foot, which represents 20000 feet on the ground plan ( °g^^Q 2OOO0 )• Positive bromide paper is employed and the en- largement made with a copying lens, in order to secure good detail. The prints are developed in the u^ual manner, washed and^ dried and are then classified in scrap books, with pages properly numbered. The essential condition for a correct enlarge- ment is that the negative be parallel to the easel or board on which the sensitized paper is stretched. The apparatus should be put up so as to fulfil this condition: if not,it ma# be reali2e<4 as follows :- -—^Expose a plate tq light and develop It complete- ~~ ly dark. With a fine point, draw a square or rectangle tt x ^rlm iSil X ENLARGING ' 245 on the film and introduce the plate in the enlarging camera. The image received on the easel should be ^x another square or a rectangle similar to the one de- scribed on the film; the direction an The displacement of the easeO^&hould be; - Various devices have been proposed to correct the errors due to the distortion of the paper in printing: thdy \ riy^^crehsTsT^ f n~tTie rmpressibn of a network of squares on the imago. P 2 % . ' T^ 248 PHOTOGRAPHIC OPERATIONS - \ The inrpression can be made on the negative it- self; for instance a ruled glass plate may be intro- duced in the holder over the sensitized plate, the ruled lines being photographed with the subject. It would be very inconvenient' to carry out such an arrangement; provided 'a small stop be eraployedj the same result is attained by means of a net fixed to the camera immediately ^n front of the holder. Hair has been proposed for the network, but it is too much affected by moisture and is too liable to break to be of much service. A rulf^ glass platfe is probably pre- ferable. The lines may also be impressed either before or after the^'xposure on the subject, an additional exposure being given under a ruled plate having c^ear lines ruled on a non-actinic background, such as collodion mixed with aurine. The lines on the print will show white. // Instead of having the network on the negative, it may ,^$e impressed^ on thiB print by exposure under a ^■ #*^ tr^ "to^ the last / ^ y \ :W^- '■ J V f ':■ ^€ .Jl^^. ■<' m '«'< i« ' - '" ' '' mv ^ mmv ft. r ^WR , :<'M,^ 1)1^' ;'«■- Mi''* % ' Mi . 1 -I r V # ■ -^ ^^ - ' CHAPTER VI , V._., FIELD WORK 115. TRI ANGULATION. The tri angulation may be ex- ecuted at the same time as the topographical auryeyi but it i% pfVf6rable to have some of the principal points located in advance by a primary tri angulation. fc The subject is tully treated in the standard workg- j«j.*iwrvej|||^i^^ry kittle requires to be added ^f» herei; However, tnfere exists some misconception as to % ■ -^ f. ' ' . % " ■- '■ ■ ■ . . ■■ . the order to be followe(Fin the.ope^^ttonsia few words of explanation ^ay provd^ useful. ' Y 'f* " % ' -^ "' ■ A survey mustV be co^idere4» tfs consisting of____ / ^ ,jf y. fet operations. ^nr^i4s.^r obje'Ct the n of the shapyfe at form of the groiind, \ 252 , . FIELD WORK Son the other the determination of its absolute dimensions^ A perfect plan or tri angulation can be made without •"S^ the measure of any base or length: the plan will ex- hibit the various features of the ground in their exact jjroportionSf but no absolute dimension can be measured on it imtil the scale of the plan has been determined. This is done by measuring on the ground one of the dimensions represented on the plani'^o the object of the^, measure of a ba8,e is to fix the scale ■ of the survey. To execute a triangulation, the surveyor is recbm- mended to commence by raeasriring a base and to make it the side of a triangle, on which he will build others of increasing dimensions. There is a certain logical sequence in the order followed, but in strict thdory, the order is inroaterial, the triangulation may be ex- ecuted first and when completed, connected with a base \ by triangles decreasing in size as they come rtdar the base. In practice, the case is different: there are several advan ta ged in e x ecu ting the triangulation, before the measure of the base. < I . i / ■0 ■ *^ TRIANGULATION 253 AH, .. f The choice of a base must fulfil several con- ditions: the ground must be tolerably level and free of obstacles, and the direction, length and position of the base must be such as 'to permi.t a good connec- tion by triangles of proper shape.wlth the main tri- anguli The surveyor can make a better choice "0 after he has been over, the whole ground than on his arrival, when he has seen little of it. Having estab- lished the main triangles, he will also know besiteto . connect them with a base.. In a mountainous country the principal summits of the trian^lation are fixed by nature and cannbt be changed while the position or direction of a base may generally be modified to some extent. Were the base measured first, it might be found not to connect properly with the raai angles, y - ,; ' y The secondary triangulation is the work of the ^ topographo^fcand the construction of signals on the secondarjnpoihts should be his first iict on arriving on the ground. — V — "YlOt Shol^d the time iat hiS|^isposal allow, he will lomnence the survey proper'^ntillall signals : I /^ ^ 254 FIELD have been as tabli shed, ciHiiM|p^emky have to meaaure^ngles betwee«pi?olnta not veT»y »eli definea^ In such a case, the closing error of a triangle is as- sumed to be due %^ the want of def^itfon'of th^ ':^' ¥^ points. Let A,B an#^ represent the angles of a tri- angle, whose sffiits have been occupied in. th« order given. At A, the sui*^eyor observes the angle between [ B and C, where there aie no signals. He puts up a, ft* signal at A ^d raoves-^o B. In measuring the angle between A and^ c, he has f signal at A and none at C. Placing a signal at B,he mea.sur^, ||e third. angle C between two signals. / Call oi the closing |rror of the triangle and e the probable el^ror.of a dig*^!^ a po signal. The' probable errors of the ^n r Tor A eVF f ■'% % ^ 4, t' The corrections' applied to the angles must be pro- aortio r le-orr zW^> iiiBiiWtaTa ilM««(»«(ai»iw««T CAMERA STATIONS 255 J- n for B. OCr '%, t* ^ 77yr~. .The closing error must not exceed a certain limit fixed by the degree of precision of the surveyrwhen the limit is exceeded, th© stations must be rl-oc- cupied,coninencing at themost doubtful one. IJhe stations of the primary triangulation are the last ones to be ogcupied when they have been ea- tdtbli sheet by a previous survey. , ^lo have a correct idea of the work he is doing, ,Q Jirveyor must make in the field a rough plot of I jur hi 8 ^Wl angulation, on which he marks all the stations qccupied. It will show him the weak point's of the survey and pemit him to plan his operaii|^ with more assurance* ^ f^^^ The object of the secondary triangulation is to fix t^^e camera stations-: itti summits must be selected for that purpose only. All the /topographical details of the plan are drawn from the camera stations. 116. CAMERA STATIONS. A camera station is fixed either by angles taken from the station on the tri- angulation points or by angles taken from the latter 31 :'■■?. J' » ^y I ■: 256 V FIELD WORK or by both. It is more easy and more accurate to plot a station by means of angles taken from /the tri- angulation points than by the angles measured at the . station, therefore the camera atatibns should if pos- ■ slble.be occupied before the triangulation summits: , ' there are, however, other considerations which may prevent it. - Camera stations must be chosen in .Viisv^ of the construction of the plan by thQ method of i^itersec- if- tlons: other methods are to be employed only when this one fails or when the data collected on the ground are insufficient to fuimish a sufficient number of intersections. A raiark or signal of some kind should be loft at each station; it does not require to be very elaborate, a pole or a few stones are sufficient. Angles on this signal are measured from the triangulation points, in order to place the station on the plan. I ■ It will seldom happen that the d^m^ra is set up precisely at a triangulation point. Generally it will be advisable to move a few feet in one direction or another, for including in the view a certain part of --/- E^'Sfcl. jtAtJti^Hw .„;■?■ \>,v.\l made ofi thp spot. ^^Vie sketches permit to identify ir points with more certaint.y than a mere designa- ■"'■■■■'. ♦ ■ : t ion. by a letter or fi^re. •„ • ■'^#,' • ''■'■' 'V.'" :*^- ^te= . /■ ■^M S*t ^. ■&^". \^\ w. -I? A I- f ;, asBs • ■■•S'^''. -;■ ■■■^' «■ 1 i V '6* - • CHAPTER VII •<4' \ PLOTTING THB SURVEY 117.' SCALE OP PLAN. „.The minutes of the Canadian Surveys are plotted on a scale of 20600 '» *hey are afterwards reduced for publication to '2 .ac\(\c\ • '^^^ 40000 equidistance is 200 feet. >*• The convention already adopted in Perspective ( $ 54 ) must be recalled here: the angles measured and the photographs taken must be assumed to have been measured and taken on a model of the ground already reduced to scale. That the perspectives obtained from any point of ayieh a model will be the same as tHose taken from 7 "•ill »„ SCALB 261 ^ the similar point of the ground has already been shown ( $ 54 ); the same rule applies to photographs. in theory at least. The angles measured are also the same as *bn the ground, for any triangle ABC,Pig.l49, of the ground Is represented on the model by a similar triangle a/^O- The altazimuth set in cc/ will give between h and c the same angle as it would iSetween J/ - - (' .u Pig. 149 B and C„if set at A. '&■■■. ^ * Thftia if thepldn be re- qulired on" « scale of 5(JUW the model will be assumed to he^j^ been reduced to that scale and the problem eonsiets %n making a plan ^ fyll size by m^ns of angles and^, photographs obtain- ed on the model. , i/ No Change being made to the camera* the fooal » ■■ ■ I ^i length pt*e8erveB the same value; If one foot* It will cover on the model a distance oofn^eapdnding to twenty > thousand feet on the ground* / , The p l an an d ^ he m od el being both iwdufted to the- •■•il wwii'^.^'\::V- : \ tf TRIANGULATION 263 le e .ordinary man. v,- Of course>aXl the constructions used in plotting the plan can be demonstrated^ without such an hypo- thesis, bux the explanations would not be so simple ah^ Jit wpuld not be so easy to grasp 'the whole sub- ject. 118. PLOTTING THE TRIANGULATION. The primarj*tri- ;\i angulatipn is assumed to have been previously cal- j> culated:the primary stations can therefore be plot- ted at o.xiCie by their co-rordinate|. The ajigles or the sec6ndary triangles are row calculated, and the corrections indicated by the ©losing errors, applied. Some of l|;iese triangles have common sides with the primary triangulation: tftey are calculated first. With the values found for their sides, th6 adjoining triangles are cal- . ■' . ■ ' ■ . "»-• "■• • .'•"-'■ ^, f" culated and so on, until the lengths of -all sides have ^ ■ ' ■ ■?' been obtained. •' "^ " , •■■' to ' With these values, the differences of lat^ude ' And *Jepartu>e fronr. every sonuait of the secpndary ti'i- ^ #•■' -^ angulation to the nearest primary station are cal- . . eulaied. Unless the prinmrV tyjanglee be very large, m $ « ■■'• ■■■■ . ...' ™ - ' ^ /. 1 V ^ •>* "■ IP ■ V mtm mmmlm 264 PLOTTINO THB SURVEY I ii the seoondax*y stations can be plotted on the plan by their latitudes and departures without any appreci- able error. The camera stations are next placed by the angles .o^ii^rve4 upon them from the triangulation po^^^^lheae angles are plottejl with a vernier pro- tral^P^F by means of a table of chords;|9ither me- ■^ thod is'^accurate enough for the purpose. I. Aa long as a sufficient ntanber of c^readings have^i been taken on a camera station from triangulation polntStno difficulty is experienced in placing the ^ station: it Is not so when only a limitod nvunber of readings or none at all are available* There are two cases to consider. Case I« The camera station has been observed from one or mora triangula- "' tion points. The camera ' station K Pig. 15(r, having been observed from the tri- angulation jpoint A> trl- Pig.150 V angliBs may be formed with M,A and other triangulation . TRI ANGULATION ' 265 points observ-ed both from A, and U, such as B. In the \ triangle MAB,the angles at M' an4 A hdve been observ- • " ? « ed and : B =-180<»~( A ^■M ). » Similar calculations being made, for other trlangula- tion points will give the direction -of the station as seen from these pointatthe plotting is done as if the station had been observed from every such point. ,. ■■■ ' f> Case II. The camera station has not bsen ob- seirved from any. tri angulati on , point. \ln this case the station must be placed by the angleis mioh have been observed jProm^t. This can be done eitiMrr by describing/through the points observed, circles con- •tainlng the angles between thq||^or bjTthe use of a station pointer." The first method j^qui res cooiplicat- ed constructions and is not very aoctirat^ and the station pointer can serve only for thrse points at a time. The follo#ing prooess will be foun4 rapid antf accurate When many points haVe been observed from the station^ On a piece of ;ti^acing paper« take a point \o represent the camera station and draw the directions \ ■/--'i 266 PLOTTING THE SURVEY Of all the points observed. Put \the tracing paper upon the plan and try to bring every one of the dia recti on a drawn to pass through the corresponding point of the plan. The camera station is then in its ./ "^ ' •■ ■ place.' '-* Prom the foregoing, it is clear that the surveyor should endeavour to obtain at least one direction from a triangulation point on every camdi^ra station: the plotting is less laborious and the result more ^*^ accurate. , .The use of photographs for placing camera stations must be avoided, the precision is not suf- , ficient. -■ > 119. PLOTTING THE TRACES OP THE PICTURE AND PRIN- CIPAL PLANES. The horizoh euid principal lines are X K- V ^ drawn 6n the photogr^hs through the proper points . / ■' ■ . - - \ of the comb marks (\5 108 );the hprizo^ line, however, is checked fey'the altitudes observed , In the 8S(me manner as it was placed on the first print^^hen- de- termiriing- the c'Ohstahts { § 108- ). •*►■ if .,.,'' „ „ • ,, ' . ' The traces of the principal and picture planes ■^ are now drawn' on the planv Ever/ photograph contains ■t^- v.. A / Dr V ^ TRACE. OF PI CTURB PLANE V 267 ^t l^east oiii0 andf^nerally several points of which the directions have been observed and raftj^ed on the pl^. Pind the distance jSa. ^ P^. 151, from the station to the projection ' of such a point oc n ' A • - 1 \ fl' 1 ■ 'r ".v Pig. 151 of the photograph on the horizon line; PS is taken on the principal line equal to the focal length and P« equal to- ^S^^ tem- ; • ' . / g^rit to it: it is the trace of the pirincipal planei / twee * / > The ^race of the pictuSpe ulane is the perpendicular • ^ ^ /■■;■■ ■ "-^W^ to S)p^ passing through Instead of makinff th#.e&*i«t ruction on the / s**-' • '■''i i i i i 4P r A 268 - ^ PLOTTIl^^fHB SWRVEY •»■<*" photograph boardfit can be made on the plan. On St A take 9^ B ( Pig. 152 ) eqtial to the focal length, erect -.'. . BC perpendicular to S, A and equal tp (xoo (Pig. 151). Join 1^ C and take S/p equal Pig. 152 to the focal length: at ;t? V ^ erect a perpendicular to 3^ C;it i8-,the trace of the picture plane and 3^ C is the trace of the priigRl^l plane. The first met^i^^^Wpreferable, because jLt does not require so many construction lines on the plan. The trace of the principal plane is marked onl2 where It intersects the picture trace so as not to confuse the plan. When the directions of several points of the photograph are shown on the plan, either because the directions were observed frwa the station or inver- sely because the station was observed from the points, it is better to proceed in the same manner as in '. . . 5 108 when finding the constants. Mark on the edge of a band of paper the projections of the points' \ \ TRACE OP PICTURE PLAKB 269 ts> images on the horizon line and try by mdvin^ the paper on the plan to place every mark in eoincidence with the corresponding direction of the filan;the edge of the paper will then be Vao xtrace of the picture ' jplane. The advantage of this m^ftbd is that a uniform contraction or expansion of the print does not affect the ♦accuracy of the plotting. Even with two directione only shown on the plan, it is advisable to' eajploy the same process, but the two lines not being sufficient to determine the trace of the picture plane, the edge of the paper must be kept perpendicular to the trace oflthe principal plane drawn as if there were no distortion. This is equivalent to assuming that the contraction or expansion is uniform all ovej* the print. I ■ % I The effect of distortion may also be corrected to a certain extent by modifying the focal f^ngth employed. It has been shown ( $ 114 ) that Ian enlarge- ment or reduction of the perspective is equivalent to an enlargement or reduction of the distance line. AssTuning the contraction or expansion of a print to be xiilifonn, ttie effect on t^e perspective is the same > as if a shorter or longer focal length had bebri uRed. ^^^ On m the irfre- -■' on « ■ ■*■ ■ a: " ti .1* -<: n -■ ^ *.*m 6 / » ■ i^.. • > - ■ ■ -— TT- • " -, ■"•■-!'• , '^- T / k f - ,i IMAGE EVALUATION TEST TARGET (MT-3) V- m // {/ ^ ,% <; ,,v A ^ % :<» :A fe ^ % ^ 1.0 I.I 2.2 lt% IL25 III 1.4 2.0 1.6 / .Sciences Corporation •SJ iV 5\ ^\ LV <: 23 WEST MAIN StlMl^ . WEBSTER, N.Y. 14580 (716) 872-4S03 '^ ••i* 270 PLOTTING THE SURVEY t! The correct length is found as follows. On the trial print ( § 108 ) measure the dis- tance between the two comb marks of the horizon line; call it a/ and let / be the focal length found. Then If the distance of the comb marks be measured on another photograph and found to differ from <5Z^ de- signating by a^^ this new distance* the enlargement or reduction of the new photograph compared to the trial print is ay » > and we have the proportion: a^ f' "^ ~ 7^ from which the correct focal length /"'is deduced. Should the distortion of the prints be such as to require frequent corrections to the focal length, it might be well to make a scale which applied to the measure of the distance of the comb marks, would give at once the focal length. For instance suppose rC and y to be 25,000 and 20,000 feet respectively, on the scale of the plan. Draw ^ line equal to 25,000 feot and divide it fnff^SO, 000 parts: this scale % is ■■Sfa-iiii- ' INTERSECTIONS 271 ■'V-.. % applied to the trial print for measuring the dis- tance of the comb marks, reads 20,000 which is the value of -the focal length. Applied to any other photo- graph it will similarly give the correct focal length, ^et paper expands more in the direction of its length than in the perpendicular direction.. In the 'A case of prints showing an appreciable difference in the rates of expansion or contraction measured on the horizon and principal lines, different focal lengths might be employed, one for the horizon line for plot- ting the points by intersections, the other correspond- ing to the prj.ncipal line, for measuring heights. In practice, it is found that none of these modes of correction are required when bromide prints on good heavy paper are used, provided they be all treat- ed alike and dried under similar conditions. 120. PLOTTING THE INTERSECTIONS. After drawing on the plan the traces of the principal and picture planes, the draughtsman takes two photographs covering the same ground and marks by a dot and number in red ink the corresponding points of each. The points 's.re chosen on those linos which define best the surface, I > ^ <':' t fr- ■ '^'2 PWTTINO THE SURV^y such as ridce., ravines,, treama, crests, chalices of slope etc. He marka on the edge' of a band of paper the distance of each point of one of the photographs ' from thi principal, line and adjuat. the paper on the trace of the picture plane previoualy dra.„ on the Plan.holding it by paper .elghts:he repeats the s«ne ^eration for the other photograph. Inserting' a fine needle at each station.he faaten. to it a black silk thread connected at the other end by a fine rubber bafd to a small paper weight. Holding the weight in one hand. he moves the thread until it coincides with one of the marks on th. edge of the band of ^,r corresponding to the station and he deposlvPK weight on the, plan. giving sufficient tension to the rubber to keep the thread taut. DoiKg the same thing at the other station. the intersection of the two threads indicates the position on the plan of the point of the photographs. ' ,, When tlje bands of paper overlap, as in Fig.153, the portion OD of the picture trace P« is marked on the band m which is undem.ath,the band PQ is pioja'a ih proper position and th. mark, on it, edg. trana- . . M 1, \/ / / \ " INTERSECTIONS 273 ferred to the line CD. The band PQ is now plao^ed under MN, the marks on the latter along CD sex^ing the same purpose as those of PQ. The station may be too close to the edge of the plan for plotting the trace of the picture plane* as for Instance A. Pig. 154, the picture tr#ce falling in QR, outside of the plant Tn this cas6 the trace AC of the principal plane is produce(5 to B, a distance Pig. 153 I , — 5_ \y «.' Pig. 154 eqtial to the focal length and MN is drawn perpen- dicular to BC or parallel to QJl. The line MN oc- ^ cupies with reference to QR'the .'same position as the ■e focal plane of the camera does to the picture plcme of the perspective. The direction of a point of the photograph projected in il on the picture trace, is found by joining NA and producing to the opposite \» side of A. ^'^'* « PLOTTING THE SURVEY The first two intersections ahould be checked either by a third one or otherwise. They may. fo'r. in- stance be checked by determining. tho height of the point from the. two photographs lunless correctly plotted, the two heights obtained will not agree. This check. however, does not indicate slight errors. The Check may also be a line drawn by means of - the perspectograph or .perspectoraeter and on which the point is situated such as the shoW of a lake or of a river. but the best check is a thltd intersection. /■> i The mmbQf of every point is inscribed in pencil on the plan. ■•' 121. PLOTIIigG WITH TIIR PERSPECTOGRAPH. ' To draw with the perspectograph the plan of a figure which appears on a photograph, the figure must be beyond the picture plane ( § 98 ) or below the ground line o» the photo- graph. Thus the lake AB ' (Pig. 155), being below the ground line XY of the' photo- gra pto c annat b e drwrrt w ^ tin- ^^ V Pig. 155 I V 1 " ' ■ ■ . s^ '- ' < .^ ^-- ~ PERSPiCTOGRAPH . .'^ 275 , r ■ . - f out a change of ground p;.ane,such that the new ground I ' ' - line xv'be below the lake AB. It- has been explained that this is done by doubling the height of the. ■itation until the ground line is brought into cor- rect position ( § 98 ).' ' The slide XY of the perspectographjFig. 156 is adjusted by the scales drawn in X and Y on the dfrawr- ing board, to a distance f rom 'RT equal to the focal length. ' • .!■■ . . ■ - After adjusting S I , / the pencil is brought over a point, M, of the . ^'■ trace GH of the prin- cipal plane at a dis- tance slA from s equal to twice the focal length. The photograph ;|La pinned under^the tracer. J/ xU, J o\}^ r -A'A X " I i '/ yf A- 1- ■M\ n u .«' I 1_. ff .«■•. ground lines. . «^^ '-. The oroas section^pap^r la pinned to the board, one Of its Hnes coinciding with the trace of ttt^ principal plane GH, and other lineB with the front lines AB and CD.drawr at kno'*n distances from the fa©t Of the station s. There will be. ^o difficulty in tracing wi th the point 7* the part of the photograph »rhich on the \ ^ - figure la on the right of the principal line, but it\ ' may happen that in moving ^ to the left. the obli- \ >quity Of the ^rm MS be such as to prevent the free ^ pW of the instrument. It shpuld then be reveffttd, thVslide Xy being changed end for end. the photograph transferred from EP to KI., the cross sectio^p^^^?- .,. mov^d so a^ to bring on the trace NQ of the principal Plani the line of the paper which was formerly over GH,an^ the point S- placed to the left of ^;;x beinft now beltweon the two alidea RT and XY, the tracer has to be ihanged to the opposite arm. Trie perapectograph can be so adjusted that the trace ojr the principal plane is the ««».. ^.^^th ^ PERSPECTOORAPH 277 -■»' positions of the instrument, it being sufficient not to move s ^ when inverting the arras and slide XY;^the cross section pappr then doe's not require to be dis- plaoed. Raving obtained the plan of the figure shown on the photograph, the reduction to the proper scale is made at sight on the cross section paper* and trans- ferred to the general plan. The transfer shbuld be checked by points previously established by inter- sections. The use of the instrument lis possible every time ,the plane of a figure can be determined* as for instance a lake* a river, a contour tBti^-tor the foot of a mountain. Slight differences of level do not affect the result when the height of the station is great. \, The instrument could also be used for figures jLn inclined planes such as a river with a rapid slope, the outline of a stratification plane which has not been distorted* a rotid or a railway* 122. HEIGHTS. The heights of the points fixed by inters ections a re f oand as explained i n t 85 . The 278 PLOTTING THK SURVEY distance from the point to the horizon line is taken • with a pair of compasses on the photograph, qne of the points of the compasses is placed on the division A of ttie sector, ( Pig. 157 ).0A being equal to the focal length. The sector is then opened until the other point ooinoides with the corres- ponding division B of the V\' /■f^ other arm. With the same Pig. 157 ., compasses the distance on the p.lan from the point to the picture line is taken, one leg of the compasses being placed in A on ^he sector, the other one will come somewhere in C,the compasses are then turned round on C and brought on the division D of the other arm corresponding to C. The line CD is the height of the point above or below the horizon plane. which means the height abo^e or below the station. Another method consists in making use of an angular scale^ as Pig. 158. Take S^ equal to the focal length; erect the^ perpendicular PA to SP and divide th \ < ^v*- HKlGH'fS 279 ^ both into eqtial parts. Join to S the points of d:^- vision of PA and through those of SP draw parallels to PA. Now with a pair of compkaaes, take on the photograph the distance from the ^ point of the perspet;- J tive (to the horizon line: transfer it to Pu and sup- pose Vh»t It is found to correspond to the "line Su, paasingVthrough jthe point 9 of the graduation of PA. TakeNvith the same compasses the distance on the plan frSpthe horizontal projection of the point to the picture line an4 transfer it to P,to the right or left of P according' as the point of the plan is beyond or within the ►'ipictureline. Then take with ihe compe^ses^he dist^ce on a parallel ?7Z/B to PA, between ?ny and the point M where the line mB is in- terseoted by 0p. corresponding t.o 9 of the graduati-on. This distance /nU is the height of the point above xOXL^elow the atatio n. i . - . ... ^ / { r 2 » r I 'f 280 ir- h 'p-i J Pig. 159 PLOTTING THE SUl^VBY " "* ' ■A scale is Aoi. pim„ed somewhere, pBrp«n<«cularly to , Uto AB.tha dlilston'o Of the ,e«i, correspond- ^^E to AB bei'hgf^tho height of the station. The oompaanea are taken off ;fhe sector.apd ore of (' B ^^^ leg4 being aetrXn C.t^e other leg coincides with a di- vision D of the scale. above or be,low C, which 1 3 the hefght of the point above the datum plane. This height is entered In pencil' on the plan.enclosed in a circle,to distinguish it ' fro« the nuiriber of the station. It is checked by i[: ' secondphotograph and-when th^. discrepancy between ' ' the two helghts^B Within the^in-its of error a^- • ' mlBslble, the »e.n is entered in red ink on the plan and the penelfTflgures erased.; A difference in the heights ^obtained from the ^ two photographa l^ndlcatea that the two points identi- fled do„ to XY and produce to r.gr being e,ual to^the height on the photograph of the point 'a image ab<>»e the horizon line. The lnt,?aeotion of AQ and *'r is the vertical projection ;,'of the point, letting fall the perpen- -"°"^"' ^' -^ g an a jr»4ucin«, -.t.t«»tt^,rw— position p Of. the point on the ground plan. -•■ '*' ' ■■ ) . «-»■*• PHOTOGRAPH BOARD 283 The construction ^ives not only the point on the of ground plan but also its height o-p' . This process is the best one for plotting a narrow valley between two "^ 3? high walls: it has however the disadvantage of requir- ing a complicated construction. 124. PHOTOGRAPH BOARD. So many construction lines are employed on th&^photographs that it is advisable ?e ' to have a photograph board on which part of the lines a are drawn before hand, once for all. It consists of an ordinary drawing board, cojsje r- ed with strong drawing paper. Two linea at right angles, DD'and SS'^Fig.IBl, represent the horizon and e principal lines ;PD,PD' PS and PS' are each equal to the focal length, so .that D.D', S and S' are the left, •> right, lower and upper distance points respectively. The photograph is pinned in the centre of the board, the principal line coinciding with SS and the horizon line with DD. Pour s#elos, forming the sides of a square OTVZ,are drawn in the centre, the side of the square being a little larger than the length of "^ -pnotograpn. — • They answer various purposes as, for 'instance. . ■ » \ ^* PLOTTING THE SURVEY drawing parallels to the horizon or principal lines ^y laying a straight edge on the corresponding graduJ atlons Of the scale or marking the ground lino by Joining the graduations of the vertical scales re- presenting the height Of the station. At a suitable distance from the distance point D a perpendicular C^R is drawn on .hich are .arked by «^eans of a table of tangents, the angles fomed with , . ' J>ft by lines j y/p l^~.~-- _^/r ; drawn from D. This scale Is employed for measuring the altitudes or azlrauthal angles of points of the photo- graph as will be exDlairiAH io + «» , t explained later. on { 5 125). From S as a center with SP n« >.o^4«~ witn i>y as radius, an arc of circle PL ia described and divid*»H <«+« ° Qivided Into equal parte, fhrough '•ifi.161 tho point. Of division, and between PL wT^DrUne, ;■_— 'V'jjjft.' t . \ TRACES OF A PLANS 285 are drawn converging to S. Parallels MN to the prin- cipal line are also drawn sufXiciently close to- gether. All these lines are used in connection with the scale of degrees and minutes QR. The studs of the centrollneads are fixed in A, B,C and E;the lines AB euid -Cfi, joining their centres and those required for adjusting the centrollneads are drawn and used as explained in § 96. A square FGKH is constructed on the four dis- tance points. ^ 125. CONSTRUCTION OP THE TRACES OP A PIOURE'S PLANE. When a figure is in an inclined plane^it is necessary to have the traces of the plahe on the principal and picture planes for using a perspective instrument on the photograph. Two cases are met with in praetiee:the plane is given by the line of greatest slope or by three points* Case I. The "line of greatest slope otay be an incllne'd road or the middle of a straight valley in ^^oB a rIveF¥low8 wrtlTa rapf^T cturrent. On the pliui, this line la represented by a llhe cvb, Pig.l62»tha \i: ^(r -,A' 1J\ U-|_ :vA'\ p 2S6 PLOTTING THE SURVEY altitude of a. being 'known. Pin the photograph to the board and take for ground plane the plane of «.: draw the ground line XY. On the plan draw a.0 perpendicular to «^ and produce ii'^untll it \v^ intersects the prin- cipal line S^, and picture trace X^ Y . On the photograph take /?E equal Xo p,h , at E erect a perpen- dicular to. XY and pro- duce it to the inter- section j^ with the perspective of the line of greatest slope. ^ Take /^N equal to ;2jO and join N^:it is the trace Of, the required plane on the picture plane. Take ^% equal to ;^L and Join MQ;it is the Jtrace^ jhe^q^lred^^x^^M^i-^^^ pia »e> -^ supposed to be revolved around SS' on the picture L A 1 Pig. 182 "^1 TRACES OF A PLANE 287 XY. )h plane, the station falling in D. Produce MQ to R:DR , l3 the vertical distance of the station above the A plane YMj3 . The new horizon and ground lines are now drawn as in § 82 , ° * Case II. Take for ground plane the plane con- taining one of the points, '^Z'^ for instance (Pig. 163), and draw the ground line XY on the photograph. ' Join Qy on the plan to the two remaining points and pro- duce to the intersections B and P with the picture trace. \ " Take pK equal to ^E and erect KL perpendicular , to XY:join the per- ^-^G \'-V / a.'i }■ spectives oc and J3 of the points shown in a^ and h on the plan and \ produce to the inter- section with KL. Take pi equal to ^P, erect TN perpendicular to XY and produce to the in- Pig.163 tsTgeic t ton N wi th" the line joining the per- ♦♦x, PLOTTING THE SURVEY ^ - ^pectives oc and ^ . on the picture plane. . «3 .n. take ;,Q ecual to ^,„. Th. u„e «, ^^ the trace c, the „,ui„a plane on the prlnciWl plane Buppoaea „voa,e. around SS'on the picture pWe.the Btation bein« i„ n. Here aleo.BR u the vertleax , . height. 0, the station ahove the plane of the thr«, «l«n points, the ne. horizon ana ground line, are constructed as previously explained. lae. CONTOUR LINES. a sufft^i: . * sufficient number of heights having been dete^ined. the contour lines are dra«. by estis«tlo„ between the points established, m a rolling country, a limited number of points would pe^ -U to dra, the contour lines with precision but in « rocy region the inflexions of the surface are so abrupt and frequent that it is utt.^i, 4 n 18 utterly impossibl* to pl*t enough points to rcnrsn.^* tv. o represent Jhe surface accurate- ly The Photographs are of great assistance to the draughtsnu^n, having t he, under hi. .ye.h. is ,H,. to __„ /» modify his curves so ao ♦« ^es so as to represent the least in- , / /» CONlOiriR LINES ^9 equalities of the groiuid. Instead of drawing tlio contour lines at once on the planithe draught sman may commence by sketching them on the photograph in the. same way as he would on the plan. Every point plottea has been marked on the photograph and the altitudes may be taken t'-^om the plan. By adopting this course, he is able to follow very closely the inequalities of the surface. The curves serve to guide the draughtsman in drawing / those of the plan or they may, be transferred bjP" the perspectograph or the perspectometer. As long as a sufficient number of points is ob- tained by intersectionsf tlier© Is no difficulty in drawjnsc the contour lines, but it may happen in a rapid survey, that the points are too f^ew and too far apart for defining the surface. It is then necessary to resort to less accurate methodp. A mountain ridge which appears in cx/^' on a - photograph ( Pig. 164 ) can be divided by the contour planes, by assuming that it is contained in a vertical P^Qy^.e.' '^'he cona t ruot i on.which has been explalngd in_ 5 62 is carried out as foilows:- * V. a' 290 PLOTTTNG THE SURVjEY On tho plan produce the projection ui' of the ridge, to the Intersection P with the picture trace and draw throu/i^h the station S, C parallel ,to a^. Having pinned the photograph to the photo- g]faph board, take- from th« principal point, on the horizon line PV eqtial to p^Z and PG equal to ^F. At G, place the scale of ;|es perpendicular to the horizon line, the 3orreapondlng to the height of the station, and join th3 marks of the scale to the vanishing point V, -Havlng| now tke points of intersection of the ridge by trie contour planes, their distances from the principal line are marked on the edge of a band of JP5Pg!L^,^ /i^t^y A^_r»c tion a p i otted-ia^the usual way^ produced, to (xj give the inter- lg.l64 . equldistanc division G CONTOUR LINES • sections of the contour lines. 291 "^ - When the moTintain has roiinded forms and no well defined ridge, the vis- * /; / \«: -V, / •s; \i Pig. 165 ible outline must be assumed to be contain- ed in a vertical plane perpendicular to the direction of the middle of the ridge. The con- struct ion is made by drawing, on the photo- graph board, SV perpen- dicular to the direction SM of the middle of the outline ( Pig. 165 ). On the plan, p^hL, is taken equal to PM and from the projec- tion (Z of the sumnit of the mountain, a perpendicular » «^ is let fall on ^ M, which represents the projec- tion of the visible outline: it is produced to the intersection N with the picture trace, PQ is taken equal to /7/N and the scale of ecfuidiatance placed at =Q perpendicular^^^ the horizw^line* The points o f -^ division are joined to V, produced to o0 and the. 3 2 i V. 292 PLOTXINO TlIE SURVEY plotting done as in th* preooding^ase, or the direc- tions of the interaeotione of aj3 by the cbntour planes may simply be plptted and the contour lines drawn tangent to these directions. The horizon line contains the perspectives of all the points at the helf^ht of the station.'it is the perspective of 'a contour line when the height is that •of a contour piano. . ' Full details on the plotting of contour lines being^given in the text books on surveying, it is not necessary to repeat them here. The main point is to understand thoroughly the mode of formation of the surface and its variations under different clrcum- atatncesithe surveyor should pay particular attention to the subject, making a special study of it. Without this knowledge, the proper^epresentation of the ground would require, the plotting of a very lar^-e « number of points. • 127. PHOTOG^H PROTRACTOR. The angle between'a point of the photograRh and the principal and hori- =,^^XlJaes,that la^^he. altitude wwirnuthalahgieViF ^ sometimes wanted. he lat ^ r / PHOTOGRAPH PROTRACTOR 293 ■ ■ \ The azlrauthal angle Is obtained at once on the * photograph Jjoard by joining the station S,Pig.ie6,to the projection, a, of the point on the horizon line. If required in degrees and minutes, the distance P« is transferred to the principal * line In PG;D is joined to G t-> n fh /• '~^:-np\>. 'i-p Pig. 166 if> and produced to the scale of degrees alR minutes BC where the graduat ffSfJSK in- , dlcates the value of the azlrauthal angle. Were many such armies to be measured, the hof- izcfntal scales TV and OZ /( Pig. 160 ) might be divid- ed into degrees and minutes by means of a table of tangents, using as radius the focal length Sid. A straight edge placed on a point of the photogr9,ph and passing through the corresponding graduations of ' ■ i I TV and OZ would at once give the azirriuthal angle , of the ipoint. - - -T—^ns ^1 1 i tud e^:g the fiiigls S ,TI g .1667 of the"" right angJe triangle having for sides Sa and «« ;■ ) ■ a.*.i."A<^«iyii£J!^a«fei •^34 PLOTTING THE 3UhYPI^"%^ To conetrtjot it, take DP equal to Scr^drav PE parallel and pquai to ^z- a: , Join.DE and produce to the scale of degre^a and minutes ffC. Thlo oonatructicn is faoilit- .ated by the lines provioubly drawn on the board. With a pair of oompasses take thedlatfince from c to, the ^ principal line, carry it from P ( Fig, 160 •) In the " direetion PD'-^d from the point 6o obtained take the distance t6" the arc ML, measuring it In the direction of the radii marked on -the board: this is the distance PP ( Pig. 166). Then with tJ^ame pair of compasaes, carry rca to PE which is done by using the par- allel lines MN of Pig. 160. The construction is now completed as already explained. A^ protractor may be constructed to measure these angles: it consists of a plata of ^ranspare^^^g-ial ^ which are lines parallel to the principPB^^^ containing the paints of sam? azimuth and curves of awe altitude. ^^^^ are, constructed by plotting the angler iif^ and drawing parallels to the principal line through tjie points of Intersection wi^h the t'..' ^ n •^V; i' -J * ,C PHOTOaRAPH PROTRACTOR 295 Denoting by // the altitude off a point x jand ' •■ taking the horizon and principal lines as axes of d§» ' . ordinatee, the equation of the curve of altitude h ipl ■ (f^^) '""^ ,fi* Thifl is an hyperbola of which the principal and hoi^ | r . . ' izon lines arp the transversa and conjugate axes and ' i ■ 7the centre 4a the prlncip^il point. One of the branch- r 63 contains the points above the horizon and the other branch the points of same altitude below the horizon* The aaymptotes are. lines intcrsetstlng at the principal point and making angles equal to A^ with the horizon line. This hyperbola is the intersection by the pic- ture plane of the cone of visual rays forming the ^ angle h^ with the horizon. The ourv^ of equal altitude may be calculated '^ by the formula of the hyper- bola or they may be plotted . by points, ri^versing the con- structlon given above for ^^"•ft n^tlTig the alt ttude or ,6 1 ^ 1' Ji !_ 1 1 •'. ~"i . /" Fig. 187 ( Fig. 166 ). The complete s 2 ♦ •» ■C" •M'.'^ « '2®6 -PLOTTING THB SURVEY pi-otract'or is ahown in Pig. 167: the angular distance between the lines depends on the degree of precision required. ^ ° The instrument/may be made, like the perspectomet- or.by drawing it on paper on a large scdle, photograph- ing and making a transparency which is bleached in '^ bichloride of mercury. -. 128. PRECISION OP THE AffiTHODOF PHOTOGRAPHIC SURVEY- ING. The precision of a survey executed by the methods exposed. When all the points are established by intersections, is the same as that of a plan plot- ted with a Very good protractor or made with the plane table. There is however this difference. the number of points plotted by photography is greater than by the other methods^' Points plotted by means of their altitude below the station are far less accurate. their positions being given by the intersection of the visual ray -*ith the grom,d plane, the angle of intersection being equal to the angle with the horizon plane or to the 3BSle of„,dep] ©f^ the point. -Wltti the camera: empUyed. embracing 60o,thls an^le is always loss than pep PRECISION OP THE SURVEY 297 30O and oven that is seldom obtained in practice, a declivity of SQo being almost a precipice. •Rierefore the Intersection is always a poor one and the un- certainty becomes considerable with points near the horizon. With perspective instruments, doing mechanically the same construction, the results are still less precise, being affected by the instrumental errors. On the other hand, it must not be forgotten that ^hen these, rpethods are employed, the ordinary topo- grapher would fall back on sketching; the results furnished by photography therefore are infinitely more precise. &^&id^& >^4eiv.ufc A ^ UK «4L^ CHAPTER VIII v>;^\* PHOTOGRAPHS ON INCLINED PLATES — L. 129. Hitherto it has been assumed that the photo- graphs used for the survey were taken on plates per- fectly vertical. There are several cases in which this condition cannot be fulfilledrthe camera may be an ordinary one, without any means of adjusting the plate, or the photographs may have been taken merely as illustrations, their employment for the oonstruc- ^^^'l *^^ *^® Pl^ being decided afterwards. There are two classes of surveys in which the pltt^w are always iirciTHed. The f i rs t are secret surveys, the views being taken with a camera concealed m iHi .iJ^'K^ Zi^ >' i ■ '^ PLOTTING DIRECTIONS 299 about the person or otherwise. The scope of these sturveys is very llraited;the photographs, being in- stantaneous, laek detail in the distance and unless .objects present great contrasts of light and shade, their Images are blurred, and confuse as soon as the distance attains a few hundred yards. Improvements in dry platea will no doubt remove this difficulty to some extent, but it will never disappear completely. Another cause of trouble is the small size of the camera and plates: the views, being instantaneous, stand very little enlargement and the measurements^ are in consequence not very accurate. The other class of surveys comprises those made from balloons. It is very doubtful whether the method will ever be found practical and prove of more than theoretical interest. It re9uir«8 the consideration of an entirely new system of survey by means of pho- tographs taken on plates placed horizontally or near- ly so. 130. PLOTTING THE DIRECTIONS OP POINTS OP THE PHO- TOGRAPHS. When the photographic plate is not vertical, the corresponding picture piano of the ? j?*4<.aW^l^|i.**iiiV>..^ A^i3Ri.tii*ii«-l<.j>*»-' - -^t ^^ « >ii^IiA ' jf»\ , 4ie"- ^ T6»^ 300 PHOTOGRAPHS ON INCLINED PLATES perspective, Which is parallel to the plate Is pierced by the vertical of the statiori. This trace is the vanishing point of all the vertical lines,.hlch having ceased to be front liyi^c rront lines, are no longer repre- sented by parallels to themselves. // ./. ,<' //L / / ! f a >/ 'S ■ -''// /'~ Let ABCD Fig. 168, be a photograph' on an Inclined plate, P being the principal point and HH'the horizon line. The perpendicular VT drawn through the prin- -s clpal point to the horizon line. is the principal line. Revolve the prin- cipal plane on the picture plan, around the principal line as an axis: the .station falls tn q «« ~ ^ in s, on a perpendicular PS, to VP, H§ being equal to tne focil length. ^°'M:Lj^AXlthe^nrat ^ ine„^l^ horizontal line from the station to the picture plane; Pig. 168 / |i ' "tv m iwwH?*3*W'wep^ffi3a«^aaaiH?.3sas ' ^Ib Mu^A^hiti . / ,,/ HEIGHTS 301 S^ V Is the revolved vertical of the station and V the vanishing point of vertical lines. Revolve now the horizon plane on the picture plane around the horizon line. The station comes in S,on the principal line produced, at a distance ^S equ^l to /f S, . To find the horizontal direction of a point /^ of the photograph. <^raw the perspective of its vertical line by joining it to V. The intersection n/ with the horizon line is the perspective of the trace in the horizon plane of the vertical of the point and S??- is its direction. Cotnparing this construction with the one for vertical plates, we see that the same methods may be employed provided ^ be used as principal point, TS^ as focal length and that every point of the photo- graph be first projected on the horizdi?^ line by join- ing it to Vi before measuring its distance from the principal line. The points such as n can be marked on a band of paper and used as in the case of vertical plates. With a plate nearly vertical, V is at «i great tLi^.^i^ak.^fb.Lixt ^ ii, tJ&^Li I ^J>a %, _ 4- i ^ ■V ' 30S PHOTOGRAPHS ON INOIINED PLATES distance frpmP,.„d the perspectives of the vertical lines have to be drawn, with the centroHnead. iSl. DKTERMINATION OF HEIGHTS. , let>,. Pig.ies be on the groundplan the point seen at ^ on the photo- graph. Project on the principal plane the trfangle fomod by the Visual ray., its projection on the hor- izon wd the line n^ . on the revolved principal plan., the projection of the visual ray is S,n.',' ^„' . being perpendicular to V^.; the projection of ^ is , Which la revolved to' G anc, the perpendicular OK to.s;r 1" the projection of the vertical, of the point or its height above. the horizon plane. various device, nay be imagined for constructing expeditiously the heights of a number of points. * 132. DETEM4INATI0N OF THB HORIZON LINE AND VANISHING POINT 0^ VERTICALS. In Jrder to Ma*e use of a pho- .tosraph for plotting t(ie plan, the horizon and prln- . cipal lln.. and the vanl.hlng point of verticals .ust be inarked on the photograph. It is asmmed tha» the 'c«,era Is available ' . either before or after ^^ .^^.^rf^,^^^^^ upon and that the focal le ngth and principal point »'iVA-a'4tA. ir'A.-^i-i^'^s.j^^j'^t^Xii HORIZON LINE 303 raay be determined by the usual methods, wi th the piste vertical. If the zenith dlstamcea of several points of the photograph have beeir^^Faerved/wi th a survey- ing instrumen.t, the determination of the horizon line presents no difficulty. Aaeump a vajiishlnii; point of Vertical lines V.Pig. 169, and join it to a point u of the photograph of which . . ' ' the zenith distance Is known. Through the prin- cipal point, draw PE per- I pendicular to Vit and \ . PS perpendicular to PE , and equal to the focal Pig. 169 length. Erect EG per- pendicular to ES,take - EP equal to.E|it join SP and make the cmgle FSG equal to the altitude of U; FG is the distance measured bn Vu from u to the hor- izon line. . ] j ^ Taking an/ equal to PG determines one point ri/ // of horizon line. A eimiljir construction repeated o^ another point of the photograph will furnish a second 'A. , h -, V ,*„ av,^ ,''f' ^ \ ji t^\i. *«i"*;:%d.4-. 304 PHOTOGRAPHS ON INCH rJKD PLATES point of the horizon line. Thia first result wil] probably be inaccurate because the position of the vanishing point V is only approximate.. A new vanishing point must therefore be fixed by means of the horizon line Just obtained.and the construction explained above is repeated. The second horizon line found will likely be sufficiently precisejif not the construction must be made a third ^timo. In secret eurve'ys.meaaured angles are seldom available. but It Is easy to devise a, attachment of the same kind as some hand levels.whioh .111 mark the horizon line on the plate when the photograph is taken. failing this, the horizon line must be furnished by the subject. When the view includes buildings. the vanishing point of verticals is given at once by pro- r duoing to their intersection *he vertical lines of the. p. ,, ' b^iil^^ngs. This point V,Pig.i70 ie joined to the principal _ V°^plili§MlB^Aa made perp e n // // Fig. 170 dicular to VP ar,d QJ t iJ-'liit^u ■t^^i^ •imffi^lmmmmmmii i-mmr •4*iiiiita >• 306 ' PHOTOGRAPHS ON INCLINKD PLATES " "' • . Take an experlmetital nocative with the field camera^the plate being vertical ;dra>7 or it the hor- Izon and principal linos, place it m the holder of the copying cwr^ra. and mark the points of the holder correspondirig to the hofizon and principal lines. After inserting the 'Loldor. the camera is mo t*£^ until the plate CD i"^ .V,erti '/I it' i T ■ I al and fixed in that position. The screen AB ]k no* "adjusted at th«» M> Pig, 171 »roper distance. par- : ai^el to the plate rand i. '. the' projecte'^1 images of thQ hoi:*izon and prinGijpal lines are marked on it in such a mannei" that the mark3 will \ . / ' ! ^ appear on the prints. \y */ For copying a necative taken in an inclined position, the horizon and principal lines are drawn on it, also a parallel to the horizon thiou^h the "^1^ principal point . Th^e rtSgat tve ^ s placeW ITT the holder with the principal line or the proper marks I- «■ I TPAMSFKR TO A p^'TlCAL PLANE 307 avi the .hori /oi't ;l^ line oT the ' principal point on the iv.rKs correi'ponJ i rig ^o the horizon line of the experii.;ental plate- The camera is moved up or down until the imago of the negative's horizon line T co- incides with the horizon Tine ^ previously marked on the screen: in this position the perspective is the same as it would have been on a vertical picture plane.. For, the inclination of the camera'" is the same as when the negative was taken: any point N of the latter would have photographed in N, pn a vertical 'v ■ plate and given the same image M on the screen. With a lens of sufficiently long focus and pho- tographs taken nearly vert ic^, as is generally the case, the displacement of the camera will be too iamall to affect the definition on the screen.- \ • • » ' - The holder must be provided with means of ad- justing the negatlvejthe principal point must always occupy the same position, the plate pivoting around it. i The horizon stjnd principal lisies are indicated on. "^tue print by the liai-k¥ f i^e^ W the sci^W: the p clpal point has been displaced in copying and is now T 2 \ 1 ' V n-'- 308 PHOTOGRAPHS ON INCLINED PLATES on the horizon line. The change of picture plane can also be effected with the perapectograph.but \he use of the instrument I is not to, be recommended when the change can be made 80 simply by the photographic process. 134. PHOTOGRAPHS ON HORIZONTAL PLATES. Photographs on- horizontal plates might be obtained by an arrange- ment similar to the one described in 5 99, with a pin- hole stop In the lens ; they are^ also taken from a balloon with an ordinary camera, but the plates are Only approximately horizontal. The picture and ground planes being parallel, the figures of one are similar to those of the other: thus the photograph (xji Fig, 172 of a lake AB is also its plan and only requires to .be reduced to the proper scale. The reduction is / ■'< 'i^^' ^ / I Plg.A72 given by the proportion between t)\e distances Zs ^and^ SP from the station td^-th©^ „ wd picttirv — ' planes. When the height of the station and the focal "XT ' <-o7^ ,!;,j» sui.TiUj^^ ,t .itfi-j" » f^,*^ * i^^fi "T^ mmmmm "•"•"t^*"'*^"^ HQRIZONTAL PLATES _^., 0. 309 length are equal, the photograph is a full size plan/ To plot the dirQCtiona of the various points, the principal point P 6"f the photograph is placed on the foot of the station s, and a line of known direction, such as Pa, on the corresponding line of the plan ^A. To find the direction of any other point B.its perspective ,4* is joined, to the principal point P: this line coincides with * B on the plan. The height of a point is found hy taking SP.Fig. /■ 173, equal to the focal length and Ss equal to the height ,/ . Fig. 173 ^ Of the Stat Ion, drawipg Pcc i .^- ■ ''' "*■ and Sf.t perpendicular to SP.Pa heing equal to the distance of the point's per- spective from the principal point and So. equal to the distance on the plan from the station to the point. J~oin S^; the parallel cuk to SP is the height of the point above the ground plane. * Photograpli taken from gt balloon cannbt"be~pe^ 77 fectly horizontal; to make use of it for plotting the X ,aii»i>,-,l' m •^,i< .1" ititi ■«"■ 51 ' J/ iT ,1 :#;;,,>'.. <), • «'.* c> .i^- 'v'lP' ! >li ii 1 1 ■ ii il 'i 310 PHOTOGRAPHS ON INCLINED PLATES plan, the trace 5> Fig. 174, of the vertical of the station on the picture plane must be known. The directions of the principal line si and of the perpendicular to it^AB, A, |/' // A- '(' J -^^ // are the same on the plan and on the photograph; they are differrnt for all other lines. Pig. 174 To find the direction on the ground plan of a point ^ of the perspective, -iraw PS perpendicular to the principal line and equal to the focal length, j-pin Ss and take SO equal to the distance f rom u, to AB. Draw ^ A and CD parallel to Vs and take Au' equal to SD; sfji' is the direction of the point on the ground plan, for A|U forms with ,i ts horizontal projection a right angle triangle in which the angle A .is the inclination of the plate to the horizon, which triangle is construe tea in SCD, therefore A^; which is made equal to SD is the horizontal projec- tlon of Alt, ^^ is the trace, on the lyround planft.nf the vertical of ji and the vertical plane passi ng i r ,1 WfKf'i'mmiimmm mmmmmffmiim HORIZONTAL PLATES 311 ,■} through s and the point w. of the photograph must cut the ground plane along sii- A much better way to employ these photographs would be to restore them to a horizontal plane in printing, by the process of § 131, using Ps and AB in the same manner as the principal and horizon lines of the vertical photograph. The great difficulty in balloon surveying will be to determine the trace of the vertical of the station on the picture plane, or the foot of the station on the ground plan. The oscinationa of the balloon prevent the use of any kind of level inside of the camera and instrumental measurements of angles are open to thp same objection. The angles might, however be measured by two observers located on the ground. In a view containing .vertical lines, their van- ishing point gives at once the trace of the vertical of the station; for a photograph taken a short dis- tance above buildi'ngs, this mode of determining the trace would be very good. Balloon surveying would only be adapted to 1 Vo ^ t ^Vi '"'"f -c£i,ifc,**«-*%ir' '.fSix^Js^^^htsi tt . .\ ^A^'^'k^ ''■liiiiiffw*^^ 312 PHOTOGRAPHS ON INCLINED PLATES military purposes^ altho\j^h the advocates of the pro- ,p ■ 0683 are confident that it will eventually take the place of all other surveying methods. ^ I . -j^.^ I '.' i ■' ' ' < 'm mmmm>mmiimmmm'i»m*m9i§^^ 313 3ro- the CONTENTS Page Preface , 5 Bibliography of Photographic Surveying 11 CHAPTER I. DESCRIPTIVE GEOMETRY ^ 1. Definitions, planes of projection 13 2. GFotmci line , 13 3. Representation of a point' ' . . . . 13 4. Representation of k straight line . . . ... 17 5. Through a given point, draw a parallel to a given line . ^ . , 23 6. Representation of a plane .24 7. Line contained in a plane 27 8. Point in a plane. ......... ^ .... '28 JL*-J^'^*^"g^ ^_P°^?^*^£ jj'^,^^ plane parallel to another plane* 29 • U M i.! JU^M y ■■MiM«l|W 33^4 • CONl'ENTS * . c * Page . *» '■ ' T 29 10. "Lino perpendicular to a plane 11. Revolving a plane upon one of the planes of . 31 projection ' 1 ' . . 40 12. Intersection of two planes 13o The interBecting planes are both parallel to ... 41 the ground line . 14. The intersecting planes cut the ground line at the same point '15/ Intersections of two planes. one of which is ^ . t. . horizontal or parallel to the vertical \ . ' . 4^ plane ..••••• • • • 3.6. Planes perpendicular to one of the planes of 44 projections , 17. Intersection of a line and a plane . * ... 44 18. Intersection of three planes ' *^ 19. Through a point, to draw a straight line which will meet two given lines 45 • * . . 46 20 Distance of two points ....... 21. To lay off a given length on a line . - . • 47 gg. n ia tanoe from a point to a line . . . » » « '^^''''>^** CONTENTS 515 24. m stance of two parallel plaaes ...... 49 25. Distance of two straipiht lines 50 26. Angle of a line with t>io planes of pro- jection .......... 51 27. Angle of two lines . • • ^^ 28. Angles of a plane with the planes of pro- • jection : • • ' . 54 29.. Angle of two planes .. - ....••••• 55 30. Thriu(!h a given line in a plnne to draw ah- othci* plane making a certain angle with the given plane ......■•••••• 37 31. Angle of a line with a plane . . . *• • •. • '58 32. Method of rotation:;. -'° 33. Rotation of a point -58 34. Rotation of a line . ^^ 35. Rotation of a plane . .^ . 60 36. Distance of two points .62 37. Solution of spherical triangles »....= 63 38. Given three sides to find the angles . ... 64 ■i^9. Given two sides and the included angle, tp find the remaining side and angle . • • • ,66 ■ -U 2 ^^9== iii'"'-iir mmmmm CONTENTS F,-:io -;. J > 3[l >to . . se . . 88 . .. 31 . . 94 51 » Vanishing line .......-'••• 52- Liner or figures in front planes ' - • 53. Measuring ' inc>s and measuring points • 54. Reduction of a perspective to scalo . 55. To placo m perspective a point of the ground piano ■.,....'.•••• 56. To place in jerspective a line or figure of the ground plane ...... 93 57. To place in perspective a point outside of. the ground plane . 99 58. To place in perspective a line outside of ■« the ground plane . . . • 10^ 59o The distance line is an axis of ayraraetf'y of f the perspective • • • ^^^ 60. Given the heights pf two points and their perspectives, to find the vanishing point an^ trace on picture plane of the line joining the given points ... ^ ... . .101 61o To find the intersections of a vertical line ~"~~l3yar^eriea of horizontal ylaneg^-r-- — .-p. — .^.Qifc 62. To mark (^ the peBspective of any line or u »■;' '■9. ■ ''yii'^ > A> 3H content; 51 » Vanish.tng line ......••'•••••■ -'^ 52- Lines or figures in front planes ' ^ . . . .St; 53. Measuring '' inc»s and measuring points . .' • - 38 54. Reduction of a perspective to scalo . - . - 31. 55. To place: in perspective a point of the ' ground piano ;,....'..• ^^ 56.. To place in jerspective a line or fif^ure of the ground plane 93 57. To place in perspective a point outside of the ground plane 9^ 58. To place in perspective a line outside of the groiind plane . . . • 10^ 59. The distance line is an axis of ayrametl'y of f the perspective • • • ^^^ 60. Given the heights pf two points and their perspectives, to find the vanishing point an^ trace on picture plane of the line joining the given points ... ^ ... • ^lOl 61« To find the intersections of a vertical line ~tjy~ar series" of horizontal: p^lanes^r » — ' — .__— ...103- 62. To mark c^ the peKspective of; any line or u "1#:: ;*:;■ / '■"'^■■^^■■■■— -— -•— r-i-r-Tr — n — T' - I •• ^ ■ ' ■•■■■',■ """r: / 318 C0;,'TENT3 Pago curve contained in a vertical plane, tlie intersections by a aeries of horizontal planes - ..105 6.3. To mark on the perspective the Intersections of a plane, line or ci:irve by a series of horizontal planes ........... . i08 . 64. Intersections of a prism, pyramid or conic surface by a series of horizontal planes 109 65. To p3ace a point of the ground plane by moans of its perspective .no 66. To place a line on the grouiid plane by » means of its perspective m 67. To draw a figure on the ground plane by. means of its pernpective ^ , X12 68. Vanishing scale . . n,^ 69. Use of the measuring line hq 70. Precision of the method .117 fl. To determine from the perspective, the pro- jections of a point not in the ground -plane, but or which tHS^ReTpitTaFnoW^. 119 ' 72.. To construct from its perspective a figure c i mm- : — — ' — ■ ' I m S > - COKT]'!.TS . 319 "\- - --" ■ Pafre in any horizontal plane . . . .' \oq i 73. To find the traces and vanishin;^ point of a lino fjiven by its horizontal projection and perspective. ., . i?.c 74. Given the slope of a, line and the horiirontal *■ projection of one of its points, to find • the horizojital projection and traces of tiic line , . » ;,^o;5 75. To find the traces of the plane containing? }<: three given points or t;;o given lines. . .126 ~ 76. Given the line of gr-atert slope, to find the traces of the plane.. .igs 77. Change of prOund plane i29 78. To find the horizontal projection of a figure from its perspective when the figure ia contained in a plane perpen- dicular to the principal piano 130 79. To find from its p^rspective the horizontal projection of a figure in a plane per- 80. Change of. ground plane and distance line . .138 ^ • u 2 wmm 320 CONTKNTS Page 81. From the perspoctivG of a I'i/rure in any ffiven piano, to coyiBtruct the horizontal prcjection of the figure 14-2 82. Change of stationi ground and picture plane .143 83. RcfJ-'Cted image fi - . 147 84. Shadows, .149 85. Heights. . . ,152 CHAPTER III PERSPECTIVE INSTRUMENTS 36." Simplest form of perspective instrument . ,155 87. Diagraph 156 88- Camera lucida .158 89. Camera obseura .160 90. Perspectograph ....*.. .163 91. To draw the trace of the principal plane f on the drawing board , . 174 92. To find the distance from the station to a ~»r^ front line of the ground plane~^ r~. . . .175' 93. To find the distance between the two slides 177 « liHWMI«M&iMllM$l&|j|^N ^^J,.. •■ A (10>!T]'''mTS * 94. To draw the graduation for the height of the station. 95 . To draw" the horizon. ground and principal 521 Page I .173 linoa on thn porspective 96. Centrolinead . . .\. • • • 97. Perapectometer V 98. Drawing the ground plan with the. camera lucida 99. Drawing the ground plan with the camera obscura 100. Drawing the ground plan with the perspecto- graph. . . 101. Change of scale / * * 179 18*L ^ 187 191 193 194 .198 CHAPTER IV •^ FIELD INSTRUMENTS 102. Altazimuth • 103. Adjustments. ^jip- JLOAi tTrip^ .201 .203 .208 105. Use of altazimuth. . . . : . .209 B 7 ' * ' Sv#8*7*>i • ^^^^^^^^S^^^S^^SSSSSSKfl^S^II^^ 322 CONTENTS Page 106. Cameras 2II 107. Lenses ' £14 lOe. Camera of Canadian Surveys. ....... .222 ioa. Use of camera 225 110. Adjustments and detennination of constants of the camera 225 CHAPTER V PHOTOGRAPHIC OPERATIONS 111. ]3ry plates. 112^ Exposure. . /lia. Development 114. Enlarging . .254 .2^8 .24] 24^5 CHAPTER VI FIELD WORK 115. trlangulatlon . 116. Camera stations .251 .255 GONTJ-'-NTS CHAPTER VII 323 Page ^ PLOTTIN(}^ THE SURVEY "^ 117. Scale of plan ,. .260 118. Plotting the trianprulation 263 lis. Plotting the traces of the picture and principal planes. ........... .266 120. IJlottinf; the intersections . .271 d 121. Plotting with the persp^cto^traph. .... .274 122. Heights ........... 277 123. Vertical intersections ....;. .281 124. Photograph board. .-,... .283 125. Construction of the tracer of a figure's plane . . . . •* . . 285 126. Contour lines .288 127. Photograph protractor .......... .292 128. Precision of the method of photographic surveying ...........;... .296' -w- siAnf^\l4 f^ L mmm ..^.ll*.i.,LU>»V. i m ii j.j] i ^l i j(^ B! .tUffl 'j l > ,- ^ ' ' ■ ■ 324 CONTENTS CHAPTER VIII Page PHOTOGRAPHS ON INCLINED PLATES f I * 129. Obaeirvationa. ,'^. / * * *^^^ 130. Plo°t^ing the directions of points of the 299 photographs * * * .13,1, Determination of heights ^^2 '132. Determination of the horizon* line and vanishing point of verticals S02 133. Transferring the perspective to a vertical plane . ♦ . • • .:al3 4. .Photographs on horizontal plates 308 . w // : .. I i^' y ^^ mn^ ^ ;£'»>.- i..^ \ ''' / i li • . . #• • ■/ ■ /■ . ' ^ ^ • ',' ~ , *~ / .v 1 picture the «tat »*§ being Joii horlzont! ■.?;■*"!,' ,i"t^- ^m ^^^^ ' .. **■ V» 1 ' ■ i h ' .„. — .^.. .. t V iJs;.^,i.v. -w,...,...., .......... *( f' r - ■ .V . ' :^ 1- .