UN LAVlNii nlT (TIIVI-S A\ K>SA\ WiiriTiA IN ^b^ liV " F I ELD AT 5 » J..>L vSToSlu 1>E LAVALLt V »« SKBO. A« TlAL««»l> Dt iM;»;Sll«>-UEAL. M.A., < 1. ■ •• / J ElBLIOTHEQUt I _ « I:*!-LABBE VERREAU ON LAV1^'G OUT CUItVES '. AN ESSAY WRITTKN L\ 18G8 • ■ • ' ' ' * . • » t • • f " FIELDAT " [i**^k *5TOMO urn LAVALLE Y HOMKliO, A( TUAL CULM' 1)K IMIKMIO-KKAL, MA., C B , ETC.] Vs « « • ■ « « • t I « • • • • » • • * « • • • * •^ • # . • t « r « > t B. Q. R. NO.s. vz .f \ f hXrLANAlIoSS, 1. Eijuivalent curves are sach as ^^lbtenll sqiial central angles. 2. Tliese anirles at the centre t'.rm what i> called the d'Tfee of eurvdff're^ or f*<»lfly th<* r^ir^jt^ir' . 3. CttrieHiX'hdluij poitds iu il'J'^fient c»ircf< are an V points where the langentr, and mI course the radii as perpenJieiilar to said tangeFit? are parallel. 4. A (•(rinijound curve is coiistructed by two curves ot ditft^rent radii, said two curves turuing in the same direction and having a c.»mroon tangent at the point of meeting. 5. This point of meeting is called the pfArd of compound curoature. • 6. A reversed curve is composed of two curves turning in opposite directions and having a common tangent at .heir point of meeting. The radii uiav be the same or different. 7. Such ])oint' is named the p5e raditis equals the sum of the radii of two other curves. 60.'i98 hrju)tAi:KS. ai" 1. As all the curves described in this work circular tlie words cu,7*ve and rirele will be us(tw aynoiiyniously. tai 2. All ineaaureiii^nts are referred to the engineerpo chain ot" 100 feet, and so the vocables chain aiiwi chord mean here both the same. hy 3. Tlie point where a curve commences is t\\eq termination of a tangent, and the point in whicor the curve ends is the origin of a next tangent Therefore the terms origin (or point of curve, o point of tangencj) and termination are appliei in reference to the course of location. _ 4. When simply as often said " set up and adjiu ^ the transit " it is in the supposition that the readc:^, fully understands such instrument and is able ti ^ detect and remedy the accidents and errors whicli^ in practice so frequently occur. PEOPOSITIONS. The following are demonstrated in most works on geometry, so that it is only necessary to refer to \ them here. Fig. I. PllOPOSITIOX I. Two tangents (J A & J B) draicn to a circle from "- any given point (I) are equals and a chord (A B) pining the j)oints of tangency (A & li) forms equaV angles (I A C ifc 1 B C) with mid tangents. r> rU(>iM)SlTl()X JI. Fk. T. 11 A tangential aiujle (J A D), or the siiijiUer <»t' the '(two aii.i,^les (J A I) ^ G A D) fbrmod between any tangent (G Tv) and a chord (A D) drawn from the ipc»int of tani^ency (A), is wcafiurttl hfj half of the wmtercepted arc (^ A D-=A M or M D^ subtended hy said chord ; such tangLntial angle heing thus \eq^ial to half of the central one (-J A C I) AC M •or M I)) which the whole arc (A D) measures. \ PIIOPOSITIOX III. Fig. I. ^ The circuinferential angle (D A E), havin": its vertex (A) in the circumference of a circle, sub- tended by a chord (DE) and measured ])y half the corresponding arc (-J D E), is equal to half of the central angle (-J DOE), and therefore (see pre- ceding proposition) to the tangential one (1 A D). PROPOSITION IV. Fifr. I. Eciual chords (A D, D E, E F /' t/iecutrattin o,- at th 1 CLutre (A C J )). :oii' Q V'^.i. IMtOPOwSITION VI. rei An ai\(jle (A J \\)fiijured b>j the ineethin o/' tir tangents (J A ^ »l B) /.v eqaal to the fnippletnent ogir the cent I ill mujle (s^np. A C 15) suhtvtuLd hij tly^y chord which Joi/is' the points- of tamjeiu'U (A v.^ 1 en iiiulsaid 6Uppleiibent equah the other atujU (D J K not very properly described as that of Intersectioi of the tanijents. ai PRELIM IN xUlY PROBLEMS. SI a V ,^;.. T rUOlJLEM I. X Jg,. X. To find the d Ida) ice between two points A and B. Wlien they are accessible and visible from each other, measure it with the engineer's chain ; but if tliey cannot be seen, take several b^^arings and dis- tances, and resolve thorn as a traverse in land sur- vevin. ' If the include^ of road, a» G A mia 11 B, hy a curve. The angle of meeting of the tangent is measured, and then a radius for the curve may be assumed and the length of the tangents calculated ; or the tangents may be assumed of a practical ienglh and the radius calculated. 8 Fig. I. PROBLEM IV. ^^^ )54 To find the ra A C B does not go evenlij. (Fig. 1.) For completing or tinishing the curve, in this case, a subchord will be required, and mind that the yubtangential angle is to the tangential one as the subchord is to a whole chord. • j Gi'ven the radius C A=2865 feet, and supposing ^A I B :^ 119 ^ , then ^I xV D^l - (by problem VII), and the number of chords must be. per prob- I iin 11 Vlll, 3Ui ; that is, 30 cliurds and I subchord f 50 feet. PROBLEM X. ''o^find th' greatest radiu-s that can be used to con- ^^ ^ struct solely with it the reversed or serpentine curve A Ba, uniting G A and ga, and given . the angUs A IB (-120^) and aiB (==119 ^ ) and the distance li (—3312 fttt), 1 80^ ^ A I Br=l SU-^1 20:1-60 Jr < A B ^B CI (=-i A C B)=:30 ^ lJSO°r^aU3:=rl80^119:^6H rt^acB <.Bci (r=-|acB)=3o?.30' n. t. ^BCI (=30'^) =..57735 11. t ^ B CI (= 30^ 300 =.58905 Sum of the ii. t. t 1.16640 J^ow 1.1664.- .57735.- ; 3342; IB .57735 4-3342 1 i— 1 B ::= 3342—1654 = 1688 == i B. in the triangle J B C OB; JB;; r ; n. t. ^ B I (=30^) B ; 1654 ; ; 1 ; .57735 1654 -hi Q B _ -^ 2865. .Oiloo i 12 Again in the triangle i C c. n 1 c B / i B ; ; r ; n. t. SB c i (=30 ^ 30') c:B; i6SS ; ;1 ; .58905 16SSxl c B = = 2865 (same as above> .5S9U1 ^g PROBLEM XI. (Fig. VII.) To find two di-n'^reid ra/Jii C B and c B tohe u- in the conM ruction of a reversed curve A L wiiting the lines^ Q A and a g * given i A I angles i5 (= 120 = ) and ai B (= 130 = and distance li (= 33-^2 feet.) 180^-^ A I B=l 80^120:^60:^ ^ A C B BCI (=iACB)=30^ 180^ "^ i B=:l 80^-130^50^ ^a c B B c i (=i a c B)=25 ® n . t. ^B CI (=30 "" )=. 57735 n. t. 'Be i (—25°)=. 46631 3i Now Sum of the d. t. t 1.04366 1.04366 : . 57735 : : 3342 : I B I B = = 1850 l.L436t> I i-I B=3342— 1850=1492=1 B. 13 • n the trian C A =: l^ir'i-^ fr^ei. Vtiind the other ra^i ^ ca. In the siuiilar triangle- A a d aud C A E we L>i " Ail ; ad / ; C A ; A E (=r ^ A B 330!?; 615;; l'^65 + 2 ; A B, 6I0X 2-^5 - -2 A .1— A B:= 3305— 10rj6= 2242=aB ai5 2242 = -t:- = 1121 =aB 2 2 lu the siinihir triangles Aad and caF we have Aa ; ad ;; ca ; .F, ■ 330S; 015;; c ; el21. | SauS 4-1 1-1 ca = ; = 6030. 15 ON ORDlNAThK PROBLEM T. ,«. Y , (Fier. X. ) Giv^n the rad'rij^ G A — ^l'^(Sh fect^ and the chord B as usuol — 100, to find D E the middle it hate. C A : A E : : r : n. s. ^ ACE, •2S^j.y : 50 (= -J A B) : : 1 : n. s. ^^ A C E, - ri. s. Z A C E =-—T-~ = . 01745, Z A C E, therefore, by tabic of n. s. s. = l^*. »w r : D. cc«. Z A C E : : C A : C E, 1 : , 999-477 {= n. cos. 1^) : : 2865 : C E, . ., ^ 999^477x2865 ,..,.,. C E — ■ — = 2864.564 ; ^P=^C D— C E-=C A— C Erz.2865-2864.561=r 436. ; PROBLEM 11. . ^p.g^^ '% Jicenthe ra/Iiii^ C F = '2Se>D /eet, the middle iitat*^^ D E — ASO of a foot, and the- distance K ~ 25 T*t', V> nad tJiP (ordinate F G* 16 (Fig. X.) Ill the triangle C F H we have C F : F 11 : : r : n. :•. ^ F C H, 2S65 : 25 : : 1 : n. s. i^ F C U, „. s. Z F C 11 :=i|^ = .'•0ST3, 2»»J0 Z F C H, theiet'ore, by table of n. s. s. = 0^,, ^ Now rrn.cosZ F C II : : C F : C II, ; 1 : . 9999619 (n. cos. 30') : : 2Si5.=> : H, C H =. 99996U^^_2S6a ^ ,^^^^^ . ^ FG=.HE = CH-CE = 2S64.S9l' 2S64.564 = 327. 17 ON LA TING O CT CUR YES. PROBLE.VI I. Y\^, \'i To lay out a curve, with the transit aiul chain, ' tangential ayigles and chords. Let X A be the tangent and A the p^int of tan- mcv. Set up and adjust the transit at A ; place e index at zero and take a back^'irh: to X. Then verse the telescope and turn ^he index until it lows the tan oren rial angle I A B. and measure the ne A \> equal to one chain. Lay off the angles AC, C A D, D A E, E A F and F A G. and leasnre the lines B C, C D, D E. E F and F G ith the chain as before. B, C. D, E, F and G ill be Doints nr stationQ in th** pqrvo w^i^h vntkv ■»• ' " - >' --••^-- -«-^^^ e constructed all the way round by repeating the bove process most carefully, and bearing also in lind that should there be a subchord, as G H, tJie ibtangeutial angle G A fl must be the same j art f the tangential angle I A B that the subchori G I is of the chord A B PROBLEM U. Pig. XI To coi/iplete the tangent, that is^ after setting out 3 18 a tiutnher of statfo?is, afi />, C, D and E in curve, to finri the tatKjent KJ at the lititt siatio Prndiioe A E to K and 1 K tu .1 ; then, in^ triiiiiirlo A I E we have the aiiirle 1 A E r= I E .8 K E J — the suiii of all the tahirential angles iC the pn'CC'ding tangent N A. Now remains ] to lav oft' the anMe K E J, and tiiu tan^^ent Ec found. But supposu the curve is afterwards Ij tinned to IE, the tangent H L will bu found i laying otf the angle L II ^F = J H E = the su:.i all the timirential and suhtan;;ential angles li the preceding langent E »1 ; and should anytl ])rovunt our seeing from A further than the stai E, the curve may be continued from E, in the bJ maimer as it was commenced at A, by laying the tangential angles J E F, F E G, etc. c Fig. xii. PROBLEM III. To lay out a curvey with the transit and cl^ hy defiection angles and chords. ^ A Let N A be the tangent and A the point ol^ gency. Set up and adjust the transit at A ; ], the index at zero, and take a backsight to N. T reverse the telescope, and turn the index un: shows the tangential angle I A 1>, and measure, line A B equal to one chain, llemove afterw^ 10 ) transit to the station H i\iu\ lay off tlui (le.tlectii»?i gle O B C equal to twice the tangentiul aii^^le A B, and measure the line B C equal to one ain. Do the same at the diii'erent stations (,-, 1) id E, making each exterior angle PCD and D K equal to twice the tangential angle I A B, ;d measure each of the chords L) and l> K nal to one cliain. The points B, C, D and E will ^ i stations in the curve ; but if after setting out a mber of them it becomes exi)edient or necessary ifind atthe hist station, E for instance, the tangent I, lay ofi' simply the tangential angle It E J equal half the deflection angle O B 0, and the desired ■igent E T is found. This heing altogether the pat usual method of setting out a curve on the oiind. PROBLEM IV. Fig. XIII, 2 find the tangnitial distarice to lay out a curve th the chain and rod, or *' hj the eye " as it is lied, given the tangential angle and the radius of id curve. Let N A be the tangent and A. the point of igency ; the tangential angle O A B = l^ and the ord A B = one chain. Draw B O perpendicular K A produced in O. In-the right angled triangle B O we have 20 A B: H : : r : n. a. ^ O A B (=!<>), ; 100 : BO:: 1: .01745, i, P,,_ 1(M>4-.()1745 r 15 u — l.<4o, tlie tarii^eii- ^ ' e distance, which is known to he the sine ot'<] tanirential angle, l)ut it is used as the chord v i laying out a curve " 1)7 the eye." The deflec i • distance is taken double the tangential one, and s: deflection distance is also to be used as a chord the above jjractical rule, that, for railroad pr tice and curves of more than 300 feet radius a chords of one chain, will be found to produce, any, a very trifling error. To construct now the desired curve, from tii point of tangency A and in line with N A, measi; A O equal to one chain, and stick a pin at O. Al from A measure the chord A B equal to one chai-^ at the same time measuring with the graduated r from the pin O the tangential distance O B equal j 1.745 feet and place a stake at B. The pin at now to be removed, Next make B P equal to n^ chain, and, in a line with B A, stick a pin at ^ Also from B measure the chord B C equal to oij chain, at thesame time measuringwith the rodfro-i the pin P the deflection distance P C equal twice the tangential distance O B, thus 1.745 i 1 = 3.490. In this manner proceed to find otii . stations in the curve, viz D, E, T, etc ; and;, order to pass from the curve at F to a tangent , 21 ; M, tiilvini,' care only to moaHure the taii.^ontijil istancoT II instead of the detlection distance TG, roceed as before in this method which is consi- ered sufficiently accurate for curves on a canal or ommon road and will answer very well, if care- illy performed, for railroad curves in the 'ibsence fa transit instrument. PROBLEM V. Fig. XIV. ' To lay outlet curve comrnencing with a suhchord. Given, for instance, the tangential angle equal L^, and the radius of the curve 2S65 feet, to begin ,he curve with a suhchord 50 feet at A, point of -angency of J^ A. Set up and adjust the transit at the point A, and place the index at zero, reversing then the teles- cope and laying off the subtangential angle O A 3 ^ _. 0° 80, ) which must be the same part of the whole tange'ntial angle that the suhchord is of a whnle chord. Next lay off the whole tangeniial angles B A C, C A D, D A E, etc, and the chm-ds B C, C D, D E, etc, each equal to one chain. But should the view be obstructed at A, tlu. transit must be removed to B the end of the .ubchord A B, and sight back to A, and lay off the deflection angle P i^> Ce(iual to the subtangential and a whole tangential angle (-P> , 30'), and make the chord B ^ C equal one chain. The rest of the curve can be laid | 22 off by whole tangoiitia! aiiii^les tVoin B, viz C 1> I), B E, etc., and whole chords viz C 1), D E, en each equal one chain. It i.s true; that matheniny^^ cally the subchord does not bear the same prop.. tion to tlie whole chord that 'he subtano-enti ox anoxic does to the whole tanii^ential anole : tU error thoui^^h, arisi.Mg fiorn this supposition, is reallr^ so very trilling ui large railroad curvea wiLy chords of one chain that it is neo:lected in curves i CI more than 300 feet radius. tl T ■ * i1 Fig XV. PROBLEM VI. J To lay of the curves using long chords. ^ This is frequently convenient in preliminary t locations, and goes thus : instead of finding tluf stations B, C, D, E, F, G, etc., with tangential ] angles and single choids of one chain each, the j points C, E and G may be ascertained with a great \ deal less trouble by using twice the tangential angle. ■ taking then the chords A C, A E and A G that are , nearly t^ '.ce as long as the single chord of one cliain ; but the table of long chords contains the exact length of those required to subtend respec- tively 2, 3, 4 or 5 stations, which latter limit it is not desirable to exceed. i 28 PROBLEM VII. Fm.xvi. ), 'ff To lay out the curve when any ohstructi>t-^ hh hr^ auwM'i^ry one whdTi.siyme obstruction^ as'iH the' preceding ''pvMom^^ eosisto: For the same case mentioned we have this other i 24 method and it is performed by running a cu: parallel to the true curve, either inside or outs of it, in the following way : 1st. upon the rac A O measure A II of any convenient length aliquot part of said radius is always the best) a from the point H proceed to lay ofi" the seve stations I, J, K, L, M, with the tangential angle . and the auxiliary chord H I which is to be c culated as follows : multiply the distance A H one chain (100 feet) and divide the product the radius of the curve, and the quotient subtract Irom one chain (100 feet) will give the length the auxiliary chord if the auxiliary arc is on t inside of the true curve, but to be added to o chain (100 feet )if the auxiliary arc" is on the o side. The stations of the true curve, viz C, D, F, are easily found by laying off the distances C, L Dj K E, J F, each equal to A H. . . • • , , - • . . . • • • • • , ' • I . ' » » QuEBBC ; Printed by G. T. Cary, 1877. / K C \H Tin. 11. I A .'fr ^' T / / A G ^ K. X -^ FUj. HI. \ n .1 H Fiq. T\ TigV. I FiaYI. Fig. VII. G Fly, IX \2V/ FicjXL N i N -f^ -^^ M Fiy.XIII. JV Fi^.XIM Fia^ XV. > 'J^B Ti^.XVTL