THE TRANSITION CURVE OR CURVE OF ADJUSTMENT BY THE METHOD OF RECTANGULAR CO-ORDINATES AND BY DEFLECTION ANGLES (POLAR CO-ORDINATES) BASED ON THE FRENCH OF M. NORDLING WITH ADDITIONAL PROBLEMS BY N. B. M. AA\. SOC. C. E. (COPYRIGHT 1899 BY N. B. KBLLOGG) SAN FRANCISCO : N. B. KELLOGG, PUBLISHER, 420 CALIFORNIA STREET 1899 PRESS OF UPTON BROS. PRINTERS AND PUBLISHERS 409 MARKET ST. SAN FRANCISCO PREFACE. The chapter devoted to the cubic parabola is derived from the method introduced by M. Nordling, as explained by him in the "Annales des Fonts et Chausse*es " 1867. Translations of a portion of this article have appeared from time to time, but only so far as related to connecting a straight line with a circular curve. That portion relating to connecting circular curves of different radii, by means of the cubic parabola, has not appeared in the form given by Nordling, so far as I am aware. The formulae deduced : , in the * latter case are of general application and equally true for connecting curve with curve or curve with tangent, when proper values are introduced into the equations. That of joining a tangent with a curve is a special case where one of the radii becomes infinitely great. Some of the recent spirals, adopted as curves of adjust- ment in railroad location, easily develop from the equations of the cubic parabola by making the proper substitutions in them. Following the supposition indicated by M. Nordling i. e., considering x = L and substituting L for x in the equations of Chapter I, the formulae of (61 70), were obtained by the writer during the summer of 1884. While the theory of the transition curve as here developed is based upon that of Froude or Nordling, it is believed that some of the problems and tables, used in its application, have not appeared before. The figures used to illustrate the transition curve repre- sent spirals, as no interest attaches to the cubic parabola in this connection after it reaches its minimum radius, which occurs when its central angle becomes 24o6' ; up to that point the figures serve as well for the one as the other. OQ represents the relative position of the cubic parabola to that of the spiral having the same origin. The examples given for illustration throughout Chapters I and II, are based upon the same data, so that the results by the different methods, may be readily com- pared ; within moderate limits it will be observed the differences are small. A few simple equations of the calculus are used to derive the formulae, but a knowledge of it is not necessary for their application. The larger type may be read independently of the smaller type. The latter may be used when the curve is extended beyond ordinary limits, and greater accuracy in results is sought. By the "length of the inclined plane" (really a warped surface) the " run off" is to be understood, /'. e. the length of the transition curve. The more important formulae are in full faced type. In the arrangement of the text, I am indebted to Prof. H. I. Randall for valuable suggestions. N. B. K. SAN FRANCISCO, 1899. CONTENTS CHAPTER I. THK CUBIC PARABOLA. Section. Page 1 Fundamental Equation of the curve '! 2 Radius of Curvature 5 2 Equation of the Curve by Rectangular Co-ordinates.. . 6 3 Compound Curve 7 3 Central Angles 7 4 Offset Distance 8 5 Computing of Ordinates 12 6 Method of Deflection Angles 12 7 To determine Ordinates from Chord 14 8 To determine Chord Lengths 15 9 Length of Cubic Parabola 15 CHAPTER II. THE SPIRAL. 10 Equation of the Curve 17 10 Equation for Compound Curves 17 10 Central Angle for Compound Curves 18 10 Offset Distance 19 10 Froude's Formula 20 11 Remarks on Preparation of Tables 20 12-13 Remarks on Theory of Deflection Angles 21 14 Method by Deflection Angles 23 15 Verification of Certain Equations 24 16 Difference in Length of Spiral and Circular Arcs sub- tending same angle 26 17 Distance between Centers and Transformation to Rectangular Co-ordinates 28 18 Froude's Formula 34 18 Passing to Equations of Cubic Parabola 34 19 Remarks on Values of r, ft, and < 35 20 Remarks on Use of Transition Curves 35 21 Problem I. Semi-Tangents 36 21 External Secants 38 22 Problem II. Location of Offset /. 39 23 Problem III. Compound Curves 41 24 Problem IV. Tangents to Two Curves 44 Explanation of Tables 46 25 Laying out by Rectangular Co-ordinates 46 26 Laying out by Deflection Angles 47 Tables by Rectangular Co-ordinates 50 Tables by Deflections 55 Transition Curve in old track 60 is it tl o p C t e CHAPTER I. THEORY CF THE CUBIC PARABOLA, OR TRANSITION CURVE, AS APPLIED TO THE ALIGNMENT OF RAILROADS. \ 1. Let OG, and A,B, (Fig. i) be respectively, a tangent and ciicular cuive to be united by a cubic parabola. Let p t (the length of the inclined plane) be the distance measured on OG, and sensibly equal to OB, L,\ r t the radius of A 4 B t . From mechanics, the expression for the centrifugal force of a body moving in a curvilinear path is : */= 32. ir from which we determine the superelevation of the outer rail to produce a horizontal force due to gravity 4 TRANSITION CURVE. to neutralize this centrifugal force, this superelevation is: 32 in which r. = the radius of curvature of the path in feet ; v - the velocity of the moving body in feet per second, and g = the gauge of the track, or width of the path in feet. 32.2 = acceleration of gravity in feet per second. The transverse inclination of the track (or "cant") to produce the horizontal force due to gravity* must be acquired gradually, and if we take a distance "/" to rise one unit in height, then -4- will denote the *I,et w = the weight of a moving body con- strained to move in a curvilinear path f the centrifjgal force developed by the weight revolving about a center with a radius r. Then the expression by mechanics for the centrifugal force is Conceive the center of gravity of the weight to be moved in a direction contrary and parallel to that of the centrifugal force while its center of support remains the same until a horizontal c mponent of the force due to gravity is developed equal and opposed to the centrifugal force. Denote this component by w,\ then we by similar triangles we have w : w, = g : e\ whence w, = now by the conditions imposed we .,-/...= close approximation and the one generally used. stant. The error, however, is small in an cases wnere it is used tor superelevation. It may not be out of place to remark here, that it is the usual custom in fixing superelevation, to depress the inner rail below the grade of the center line of the track %e and elevate THE CUBIC PARABOLA. 5 the outar rail above the grade line %e t thus preserving the center line at grade. A more nearly correct value for g is The resultant of w and w t due to the effect of gravity = which is the pressure normal to the plane of the path when centrifugal force is developed. rise in a unit of distance and for any distance p the rise will be -?- = the superelevation necessary to pro- duce a horizontal force due to gravity, equal and opposed to the centrifugal force, hence : JL wh ence ,, 32.2 r, 32.2 The second member of the equation, for assumed values of v and /, is constant, and is placed equal to P, hence : r,p,= P; r,= -|~ (3) \ 2. From the calculus the expression for the radius of ds* curvature is ; r. = -77 -yr - , in which ds is the dirlerential dp, d *yl of OR, and dp, the differential of OG t (or S? t ). Fig. 2; d*y the second differential of B t G t with respect \o p t \ now as we have assumed OG t sensibl)^ equal to OB n therefore we may assume ds = dp t , which gives, thereby : di) 3 P d*v 6 r ' = j/,%: = p, inverting the terms dp? = p' (4) whence p t dp, . V' i TRANSITION CURVE. whence : t - . * <*y, = J^ integrating, y t = (6) in which _y, and p t are co-ordinates of the point B t . By the same process we may find for the point B lt of the parabola at which the radius of the circle is r llt being also the radius of the cubic parabjla at the same point and co- incident with it, i> 3 >0* = 75* which is the equation of a cubic parabola. (7) In general fop any point of the cubic parabola whose co-ordinates are x and y we will have : y 6P (3) EXAMPLE 1. Given x 150; p = 171900 to find the ordinatejy by (8) jr = = - -P = 3.27. \ 3. From the relation established in equation (3), we have : P P p tl m'.Jt . A = (9) ' THE CUBIC PARABOLA. If now we denote in Fig. 2 (10) *fi then />=to; (II) rr tt which substituted in the values for p t and ^,, give . ^^ _ ^ . ^ = ^ r -_ = _fc_.(i 2 ) ~ ' ~~ By (5 ) ;-tan..-! tan^; (13) and since the angles are small, the arcs may be taken for the tangents of the angles, thus : <^/_ _ P, 2 dyn_ __P// 2 / ryl \ EXAMPLE 2. Given # = p n = 150 ; T 3 = 171900 to find the P 2 angle a ;< subtended by L n : tan a /y = -r, (supposed = arc a. lt ) Substituting the values of ^ and P in terms of f, r, and r tt we have : a <= * r " . .. i _ ; (15) ir^t-rj 2^,^-rJ subtracting a _ a = .- = 2r, r w (r t r lt ) ir t r u (r,r lt ) whence TRANSITION CURVE. EXAMPLE 3. Given x = p = 150 ; r, = 2865 ; r tt 818.8, to find by (17): a y/ - a, = J3,, + ft = - t (1 + 1) = 75 (.00122 + .00035) - > *j rf/ 75 X .00157 a// a, = p tl -j- j8 / = = arc .11775 = 6 44' -j- p t = A,D,B t ; ^ ll = A tl D tl B ll ; if now we make alternately r t and r,,= co and, at the same time, make /3,= o and /3,, = o, we have : whence fin r n = ft i **/ y!tP > P F ~\~fi V ~P / ( 19) *The values of p, $ a y a n and <|> are expressed in arc at radius = 1. The arc of a circle which is equal to the radius in length = 57.295. The arc of 1 degree at radius = 1 is .01745 +. Hence the expression for any number of degrees in a given arc A circular arc is A = ..,.,.. , \ 4. To find the distance A.A,,^ between the two circular curves measured on a common normal passing tnrough their centers; i. e., the least distance apart of the circular curves to be joined by the cubic parabola: Referring the co-ordinates of the circle with radius r t to the same origin as the cubic parabola in which m t =.N t H t ^ we have : ; (ao) squaring the last term of the second member then, iy^i.+ir.m.+mt+r*; (21) suppressing all terms of y t and m t in which r t does not enter as a factor, since y, and m, are small compared with THE CUBIC PARABOLA. 9 r lt (the opposition of signs will still further diminish the error) and factoiing the remaining terms, we have : 2r, (ftm,) = (p. l,} 2 ; (22) in which I, is the abscissa of the center of the circle and y^p, ^'~ zr, ' &r f 6r,' whence ,or in general, m = ' br, 8r, 48^, 24^ EXAMPLE 4. Given x = 150; r = 1146 to find the length of the normal "/" common to and between the tangent and circular curve to be joined by a cubic parabola : By the general equation *3 (150)2 _ 22500 24r 24XH46 27504 In the same way we may show that [ which m it = N tl H lt (25) At \ f n To prove that the cubic parabola and circle have a common tangent at/^y,/ i. e., are tangent to each other, we differentiate their respective equations : for the circle, %- - = = - (since /, = XP.) = tan a, ; (26) off r t tr t for the cubic parabola, $7=^- = ^- (since/J = ^' ) = tan "' : (27) hence the tangents coincide. Substituting the value of f t in terms of p S> 2 r 2 S> 2 r 2 m,= P ( " rr also m lt = p ' > ( 28 ) 24^ (r t r tt ) z 24; ,A r /~~ r //) we now have m t and m tl in known terms, and by placing their values in the equations for y t and y tt in which the 10 TRANSITION CURVE. ordinates for the parabola and circles of radii r f and r tt have the same abscissa x we have for the parabola, , - , for the circles y t = ^ f +~- - ; y,,= m u -\ -- ^ 17 t 27 ^ substituting in equation for y t P = J_ " . _ t r, (r t r~ ft ) )x-trJ* ( , 2 by the same process y = ^">' < '+3C2(- < -rjT-/r,]. ( } ~ 2 (r,-^^ _ 3 [2 (n-rJr-^T+^r,,'. ( v ,_-r-r,, x .tr.-rx-r. , . - If now we substitute for x; P,-\- l Ap-p lt %P and p P t ^n j n e q ua ti ons f^ the values of y, y t and y M __ _ - ' " r 2+3 2(^-0 r/ " r/l -^ - (35) f UNIVERSITY \ ^*^*W*#*S' THE CUBIC PARABfttA. II = multiply and divide the second member of the equation by 2r tl 2-r /// -rJ^ ^// 3 ; substituting, cancel- ling and factoring, y- y - . P^,-^_ = / 2 (^-^. ( 39) " ^- 4 8r / r / Xr / -rJ 2 48r,r w By similar reduction, which is exactly the same value obtained for y y n wh ; ch shows Uiat the cubic parabola bisects /(or m) and each expression equals l / 2 f whence adding the Values yiy+yy^yuy, ; v _ v _ 7* ^- 24 if the central angle of the cubic parabola be small, not exceeding say 10. EXAMPLE 5. Given x = p = 150, r,,= 818.8, r,= 2865 to find the offset by (41), f= -i (-1 1-) =A,A,^ ( ^- (.00122 -.00035) = 22500X_00087 =0 _ 815 _ 12 TRANSITION CURVE. X* $ 5. From the general equation y ^ we see that the ordi nates vary as the cube of their abscissas. Having computed the value of any ordinate y, we have for any other ordinate y \ hence ^ (43) EXAMPLE 6. Given x = 120, x = 150, ^=1.67 to find any other ordinate y by (42;, y = = (1-25) 3X1.67 = 3.27. This will facilitate the computation of any ordinate following the first. It is to be noted that all of the above equations are approximately true only for small central angles. $6. To determine the method whereby the cubic parabola may be laid out by deflection angles, and the length of the corresponding chords, we will assume equal increments of x of finite length, (though they may be taken unequal) from which the corresponding chords (of unequal length) may be computed as well as the deflection angles. THE CUBIC PARABOLA. 13 By a previous equation EXAMPLE 7. Given x = 150 ; y = 3.26, to find a by (44); tan a = -^ = ^^ .0652 = tan 3 44' which shows that the tangent of the angle which a tangent to the curve makes with the axis of x equals the ordinate divided by */ the abscissa of the same point. If, therefore, we have as in Fig. 3, the cubic parabola ABCD, ^-^\.2,nBAB,\ (45) and ^=tana ; (46) then tan BAB, = = tan 7 ; (47) tan TI = yi " V ; (48) X-l X o tan7 2 = x'lx' ; (49) 7 7i72 are t* 16 angles the respective chords make with the axis of X: 7 2 ~7i is the angle which chord CD makes with BC. By a similar process the angle which chord AC makes with CD can be computed. For small central angles the tangents of the angles may be assumed equal the arcs so that ^'f 1 -"^"T^ = 7 2 -7i = A (50) the exterior angles between the chords. 14 TRANSITION CURVE. The several angles can be computed and tabulated, to any number which is likely to be needed, to conform to any system of "change points" determined upon aft^r X y &c. have been computed for the particular transition curve where value of P has been fixed in conformity with the character of the alignment. The foregoing method for computing deflection angles is equally applicable to the spiral. S. Given X fiO ; #i=120; X 2 = 209.65; y = .21 ; yi=\ f>7 ; ^2=8.91, to find the deflection angles for laying out a cubic parabola which successive chords make with each other -^2- = tan y =- -^- = .0035 =-- tan 12' ; ..-.- urn T1 , L-. 0245=, an 1-24- ; yi - yo = 1 21' - 12' = 1 12' ; Y2 _ yi = 4 37' 1 24' = 3 13 ; ; For long chord; -g- = tan y = A5L _ 2 * 27 , . 'iio>ii tt comparing the value of a /y and y we see y = ~. o 2 7. To determine the ordinates o, o 1 &c. From any chord as Ic, let aB o, B^b = s, AB l = iC , B^B = / , M^ x 7. Then from Fig. 3, - = tan 7, ^o s = x tan 7, J - y = ^ tan 7 -J . ~ - = cosy,o=(s -y c ) s So cos 7, or since s = = 6 45'. For r, = oo -1- = 0, /3 ; = 0, arc < = - = arc = .0915, p tl = 5 15'. r t I r,, For r n = = 0, /3 /y = 0, arc- -^8 n? 1 =' -'= * /; (68) the ordinate of the middle point of L. in which L = P { -^ ^ ) = EXAMPLE 13. Given L = 150; r /y = 818,8; r, = 2865, to find / = -- X .00087 - .815 For reverse curves the perpendicular distance between the tangents to the circular curves, produced, which are parallel to a tangent to the common origin of the transition curves is expressed by In this and all examples given on the above data, it is to be observed that the value (d) which will appear in subsequent investigation is inappreciable and need not be regarded, also that the cubic parabola and spiral are practically the same and the formulas of either may be used indiscriminately. Table (2) may be regarded without serious error as adapted to either case. If the curves r n r t are reversed, use the + sign; if in the same direction, use the - sign in equation (69), TRANSITION CURVE. Solving (69) for y^L we have which is Fronde's curve of adjustment. The above equations may be obtained by an independent method, as will be shown further on, where the values of x, y and /, respectively will appear as the first term of a series. In all the above equations if r, be made equal to infinity, the terms in which it appears become = o, the terms containing it disappear, and the resulting equations are for uniting a circular curve, with radius r in with a tangent, by means of the transition curve. If the sign of r, be changed and L tl and Z,, be substituted for L in connection with r lt and r t , the equations will be those for uniting curves reversed, by means of the transition curve, so that (61-69) inclusive, are general equations for the transition spirals and are sufficiently accurate when they are used for computing spirals for transition pur- poses only. ? 11. Tables I and 2 are prepared by the use of equations 62, 63, 64, 66 and 69, in which is made zero, except where the value of x is considerable, when 115 is used. A table should be computed consistent with the character of the alignment to which it is applied, of which table 2 may be taken as an example where in the formulas has been made = zero. It is to be observed that the superelevation of the outer rail, in the use of the transition curve, may be THE SPIRAI,. 21 made greater or less than that which has been assumed in computing the tables ; the only effect it will have is to diminish or increase the assumed value of "z ", which is equivalent to increasing or diminishing the velocity, since i and v may be made variables in the constant P t i. e., it makes the rate of rise of the outer rail to effect superelevation a little greater or less. It is, however, best to introduce the average velocity of the express or fast passenger trains in constructing the tables. Where the location is so constrained that the PT's and PC's of the circular curves are quite close together, it may be necessary to give "2 " a smaller value than would be otherwise desirable. A value of 300 or 400 is sufficient for adjustment, and good results may be obtained with a value of 200 when the radius is not greater than 573 feet. Equations (62-69) contain all the elements necessary to determine a suitable curve if intended only for the purpose of adjustment, and on account of their simplicity are recommended for ordinary use. The latter remark presupposes that Problem III will be used when the distance assunder (or /) of the two circular curves to be united is much in excess of that required for a transition curve with the fixed value of P. \ 12. If at the point B t (see Fig. 2) we imagine r, to increase till it becomes equal to infinity, the curvature of B,A f = o and the arc B t A, will be a straight line still preserving its tangency to the transition curve. The curvature at B tt will diminish to the same extent, i. e., the difference between curvature at B t and B u will be the L' 2 same as when j8, = o, and = . The ordinates x and y can be computed and laid off from the new tangent as axis of abscissa with B t as origin the same as if from O. If we now conceive this new axis of x to be curved to a radius r t the curvature of the transition curve at any point will 22 TRANSITION CURVK. be increased by the same amount and the ordinates may f without serious error, be laid off normal to the arc B t A, and establish points of the transition curve. The same reasoning will apply if r /7 =coand B lt A n becomes a tangent and values of x and y be laid off from it with B tl as origin, except that the resulting transition curve would be convex to B tl A l( . The ordinates would, however, be equal to those of the corresponding distance from B. If r u now resume its original length the curvature of the transition curve at any point will equal that of the circular curve with radius r n minus the curvature it had in a contrary direction when r lt was equal to infinity and B lt A tl a straight line. In determining $ and r = -. , data may be taken from the tables. $ 13. To conform to the case under consideration r will be measured on A t B n and y normal to A,ff t . Kquations for x and_y are equally true whether the origin be taken as O, B t , or B tl . To lay the transition curve out by the expressions for x and j, their values may be laid out simultaneously with corresponding equal chord measure- ments along the transition curve. It will be seen that the above values of x and y differ from those of the cubic parabola slightly (which attains its minimum radius when the central angle becomes = 24 6') and fulfill the conditions of a spiral whose curvature and length vary inversely as the radius, and may become infinitely great and the corresponding radius infinitely small or equal o. As this extreme is never reached in practice, its discussion is beyond the scope of this paper and will not be considered here. THK SPIRAI,. 23 | 14. The principle enunciated in the paragraph preceding the above, enable us to prepare a table of deflection angles according to the following method, which may be more convenient than that laid down in Chapter I. Fig 4- x Refering to Fig. 4. If the deflection angles from AX to any point be denoted by d 5, 5,,, it will be found by computation that any angle as DAX= 3,,= 4f = ST- (nearly), (71) in which r l4 DK. the angle which CT, a tangent common to the spiral and circular curve with radius r tlt makes with the chord AD.r* To establish the points E, F arid G by deflections at D 24 TRANSITION CURVE. from tangent DT we have, from the paragraph already referred to, EDT=8 + &, (73) and 5 the deflection from AX to Z?, and A = the deflection from DT to the point E, for the circular curve with radius r tl . In the same way GDT '= + A,,; (74) the angle DGG,, = 25,, + A,,= (a,, - a,) - (5,,+A,,). (75) If we add (8,,-f-A/,) to both members of the equation, we have : GOT = a,, - a, = = 3 + 2 A/,- (76) in which A=^A * n which A = the deflection for a unit length of DG on the circular curve with radius r in and L the units from D of any point laid off on DG. The above values are approximate. If close results are indispensable, the method indicated in connection with Fig. 3, in determining 5 5, 5,, will have to be resorted to. 15. As indicated on page 20, we now proceed to verify certain equations (3-11, 61-691. We have from the calculus: rda = dL, L = =Pr-i, (77) differentiating Pi* 2 ffr ^ ; ( 8o ) P P P( 1 1 \ a " ~ a ' = ~2^r ~ W = -2 V7~5 ~ Tjr ( 8l ) From (62), L, L,, - ; whence L L -L-P- ~ (K*\ L n -L,- p\ r ^ - ;, - - - \- v - - ;, (83) substituting in value for a n a,, we have // - *, = 4- (~- ^ -^) - ^> - PI. * &; (84) M^ + j/^ being used when curves are in the same direction and the sign when the curves are in a reversed direction. (If the sign is used the second member of (84) will still equal a.,, a/, in which a ;/ is on the opposite side of the axis of X from a,, or the curves are reversed, L = L lt + L,, a.,, = /3 y/ and a, = 0,) in which <*> - ftu + f*i (8 5 ) is the angle subtended by the arc L of the spiral as well as the sum of the circular arcs A, B, + A,, B,,, or of a circular 2 arc whose length is = L and radius r = 1 1 .* (86) ~ ~ *0 = ^- (^ -- 1 -- ); L = r<$>, in which r = some length of radii that with a central angle $ will give an arc = L or L = r(f> t L and having the same value as in equation 84. Then $ = - = 4-(-UJ-> =4-( + ), -Wing for r= / 1 ' IX 2. \ r tl r t ' r 2 ^ r lt r t ' ( -- 1 -- ) ^ *"// *"/ 7 112 -- 1 -- = -- .'. the reciprocal of r is an arithmetical mean of *n r i r the reciprocals of r,, and r,. 26 TRANSITION CURVE. If in equation (84) we make r n oc, and fi,, 0, then 4r^ : ;fc?# similarly we may find 1 -B L - jj^-ft" -r-"n,fci whence = /3//r,, T r,j8,. (87) The angles subtended by and jS, arcs of spirals in this equation, are not precisely the same as result from the investigation follow- ing. In the value L, jS,, r n and ft, r, are assumed to be equal as would appear from Nordling, and A,/ A,, the shortest distance, to be at the extremity of equal arcs A lt B n and A t B,. For large angles, this is not the case there is a difference = d ; a radius r t would not pass through the center of r n exactly. In LC the angle is divided properly. A, B, < A,, B,, (see 16.) If j3 ;/ and (3 ; be the angles subtended by the circular arcs, at radius = 1, then Lc = |8y/r// + /3/r y ; (88) $ n r n and p t r, being respectively the arcs of circles A,, B,, and A, B t . 16. It has been assumed, in the case of the cubic parabola, that the length of the arc of the transition curve does not differ sensibly from the sum of the lengths of the arcs of the two circles qual to each other with radii r,, and r,, described from the ends of the transition curve to points where their circumferences approach nearest each other and their tangents are parallel, also that the abscissas of their centers equaled % the abscissa of the point at which the circle and cubic parabola become tangent to each other and have a common radius. In case the central angle or length is considerable, as may occur in the use of the spiral, the error arising from this supposition may not be disregarded and the true value of AnB,, (see Fig. 2) and A,B, should be computed in determining the length of arc to lay off from A n or A, to establish the respective points B n and B,. The arc of the circle having the longer radius, as r n will always be the shorter. The rectangular co-ordinates of the center of the circle with radius r n as D n referred to the origin O are: OP ',= I, and H,D,= k lt and of D n , OH,, = I,,, H,,D,, = &//; whence ~^ = UH(, +&); (89) THE SPIRATv. l lt p n r,, sin a,,, I, PI r, sin a,; k lt = r,, -- m //( k, = r/ --,- w/; e ^P// >*// sin a,,) (p, r, sin a/) tan (/ : P/) , > i m N or if we have the value of f COS (a ; 0,) = (P 7 + m/) - (P,, 4- m ;/ ) p, - (r,, 4- f) from either of these equations with ^ // reduced to arc, then r,,?,, = A n B tl and denoting the difference between L&n&A n R n -\ A, B, by have Placing = -(4,;^ - A,B,) = L -&& -r,?,). ' J - L - L c . 27 (90) (92) (93) (94) (95) (96) l d" we (97) (98) (99) The point # should be marked with the name of the station at />'/ the computed length of the transition curve (as if A, B, and A tl B n were laid off in the same manner as semi-tangent of a simple circular curve). 28 TRANSITION CURVE. To determine the angle /3 (second method): Let Y = B H B, S, = the angle that the chord B, B,, makes with the axis of X, then (Fig.2a): Fig 2 a tan Y = __ y/ . The angle which a tangent to the curve at B, makes with the chord B, B,, =- l c is Y a.,. The chord B, B,, = lc = \/(y t/ y/ )2 -+. (# /y _ 3.^2. The distance Z> /7 T 7 =/ c cos (Y a,) - r,, sin (a,, a/). The distance D,V - r, [/ c sin (Y a) r /y COS (a /; a/)]. /C COS (Y a,) r n sin (a,, a y > , - [/c sin (Y - a y ) -f r,, cos (a /y - ^ />/ A, = a y 4- jS, = a,, - , transposing and changing signs, <*// - / = ^// + ^/; 17: It remains to determine the distance Z>, Z> ; , = r, - (r,, ;- /) (100) (101) To do this we must first find an expression for f (and also for .t and y) which is a line normal to both circular curves, joining the points at which they approach nearest each other; this is always on a line passing through the centers of the two circles; passing to rectan- gular co-ordinates, by the calculus, we have since, equation (13): THE SPIRAL. 29 sin < dL * * dy dL sin ; ( IO 3) rt'ic = rfA cosc/> ; in which = any angle. By trigonometry,

m cos* = 1 - -;- j-^ etc ...... 1 -^ G5 ; ( IO 6) in which m may have any value from zero to infinity, n may have any value from one to infinity and any value from zero to infinity. If in the equation = ( j j we write for convenience ( 1 \ -_ then = ~n-p\ which value in equations for sin and cos ^ give, by (103): dy _ . _ L^ / . L ^ L /. L^ L^ \ ft V 1 J2"4K^ 1920 R^ i IO 7) *It should be remembered that L is an increasing function of M and j, and for any value imagined for dL, dx or dy undergoes a corresponding change. tThis and the succeeding formula are for expressing the trigonometrical function of an arc in terms of the arc itself (see Chauvenet's Trigonometry, Chapter XIII, 1867). \a n - a, = -~ ( J = ; make r, = cc, then a, 0, " 30 TRANSITION CURVE. EXAMPLE 14. Given r,, = 818.8; r, = 2865 ; L - 150, to find . If we make r, = &>, =0, 0, = ; we have approx 150 Si " ' = \ + ) = - 91G (1 - - 001398 ' = - 24X(8l .0916 X 99866 = .091477 .-.sin = sin 5 15'. In the same way, if we make r n oo, - = 0, /3 /y = 0, we shall find that sin /3, =.02618 = sin 1 30' .'. j3 /y -f / = 5 15' f 1 30'= 6 45'. j8/, /3 y = 3 45' = the change in direction of the tangents to the extremities of the transition curve when the main curves are reversed. \ ); ( I0 8) NP.r ' integrating, ^-IFV-sijir etc ^qf) ; ( I0 9) or in terras of R, since P A L, ~rr = X r,, r. * =('-!& +") = EXAMPLE 15. (iiven r, = 2865 ; r /y = 818.8; L = 150, to find y by (110): L* {& l \ T-, T/ 2 / 1 l \- 1 > = -y- Cisr^rJ L 1 -56- (i^ -^ etc - J (.00122 00035) [1 ^ (.00122 .00035) 2 -u etc.] v - OD 99^00 ' x OOOK7 ^ = >u ^ - u 5! ^ 22500 (.00087)^ ; etc.] = 3.2625 (.9997] = 3.26225. or in terms of , since L i _ 2 ..,/, ^>^ . > THE SPIRAI,. 31 similarly, dx /' L* L* dLcoo^ll _ __.. ( 1 -fr2 384R. dx = dL (l - g~ -r 3^5 - 46( ^p7 ; etc ..... -J-J ( l , 3) integrating, fi ^ 4 ^' s '- m - - = V-405PT -'- 3456A-. 5M or in terms of , since L = 2^, also KxA^ri'i.E 16. A- = /, [l (~ ---- i-) 2 etc. (other terms)] = / 22500 X .0000007569 \. ~lcT x = 150 (.9996 :-) = 149.94. It is to be observed that the above values of x and y do not differ appreciably from the values given by (8) and (64); the difference, 32 TRANSITION CURVE. however, becomes greater as L and ( - -- j become greater. x is laid off on the arc of the circle A, B, or A,, B n (with radii r, and /) as axis of X with B, or B,, as origin, x being the abscissa of y which is laid off normal, or radial, to the arc A, B, or A,, B (l . In general, by Fig. (2), any distance m = y - R ver = y R (1 - cos f/>), ( r 20) in which R depends on -------- for its value, substituting for cosc/> rn fi p its value, remembering that /> = =-, writing / for m and reduc- Lt ing, since .)]; ( I21 ) -fete.) -)]' 1 ~ff~' etc. ) , and since - EXAMPLE 17. Given r /y = 818.8 ; r y = 2865 ; L = 150 ; to find the offset /. 1st, when the circular curves r,, and r, turn in the same direction, by (125): ,_ L ~ ( l l \ fi AIM 1 VI - J ""2T V77 ^"^ L 1 112 V r y/ f, / J X .00087 l-- . .815 THE SPIRAL. 33 '2nd, when the circular curves turn in opposite direction or are reversed. Given r,, 818.8 ; r, - 2865 ; L,, = 210 ; L, = 60 ; to find the offset between the tangents to the circular curves which are parallel to the tangent to the transition curve which passes through its origin (Fig. 9). - . .t J> - 24 |_V rn i ri ) 112 \r,,3 r r,s / -[(53.86 + 1.25) (.0000007)]; = / +/ = ^- = 2.29. If L A = L, , then - Li l ~ L ' l l If now we substitute for / its value in the equation for D, L) n we have for the distance between the centers of the two curves, D, D,, = r, [r,, ~(-L.) 1 - etc.], (la6 ) EXAMPLE 18. Given L - 150 ; r, = 2865 ; r, = 818.8 ; to find the distance between the centers of curves with radii r n and r,. 1st, if curves turn in the same direction, (Fig. 2): 2865 - [818.8 ^|^ X .0008?] ; D,, D, - 2865 - [818.8 + .815] = 2865 819.615 = 2045.385. 2nd, if curves are reversed (or turn in a contrary direction) (Fig. 9), then: If L,, = 210 ; /,, = 60 ; r, = 2865 ; r, = 818.8: X sec A' 7/ Z? y Z) ;// = [r, (/ y /y //)] sec K,,D n D nl ; ^ /),,, = [2865 (818.8 2.29) j sec K n D, D n = 8686.09 sec A',, /^ /> 7// ; tan K n D, D,,, - * Also D, D n ^ D, V sec , (by Fig. 2a^ and / r-- (r,, I), V sec ^/) 34 TRANSITION CURVE. $18. If, in the foregoing values of y, x, , and / we omit all terms in the bracket after the first, we have x = L = X\l+-j^- ( I2 8) dy_^L^= _L_ / J 1_\ = ^^ J.r 2/> 2 V r// r/ / ~ Ln r, H-oml f - 7 - ( ), and if in the value for r we write 18 V r n r, / L for L and cube it, then 6P Hence the ordinate y y^f is at the middle point of the transition curve. Many of the formula; of [ 15-17], while more nearly correct are too extended for field computation. Tables 3, 4, etc. are THE SPIRAL. 35 prepared by the use of them, and similar ones should be resorted to for operation in the field when those deduced by [10-11] are not close enough. 19. If /-/ = oo, /3, = 0, a, = 0. A, B, is a straight line = / - n S i n p fi - . r// _ t - n sin = .r,/ '"// sin a // = OH U . (Fig. 2.) 2 20, The transition curve should be no longer than required for purposes of adjustment, considering the highest speed that can be safely attained upon the circular portions of the alignment to which it is applied. The longer it is the more it increases the perplexities of the trackman, who is unable to maintain the transition curve in its proper place wholly "by eye" or by their usual methods. The aid of an engineer is indispensible if the transition curve is of considerable length as may occur when it has been used for the purposes of securing a better fitting to the ground in location, irrespective of its use as an adjustment curve. This latter contingency can usually be avoided by introducing a circular curve of intermediate radius between those to be joined, and uniting this intermediate curve with the principal ones by means of the proper transition curves based on the assumed values entering the constant P. For the above reasons, it is believed that the use of the transition curve for other purposes than adjustment should not be encouraged. With these conditions adhered to, a table can be prepared giving the ordinates at regular intervals on the long chord or along the length of the transition curve. The beginning and end of the transition curve should be marked by permanent points. A few useful problems follow in the application of the transition curve. PROBLEM I. TO FIND THE SEMI-TANGENTS. 5 21. Given a circular curve whose radius = r u \ the intersection angle = /; the semi-tangent = 7", to unite it with tangents by means of transition curves whose lengths are L t and L and offsets are / and /respectively. Fig5 CASE i. When f, > f (by Fig. 5), if /< 90, = B...A + Ac l -a l c l + b t d t - r f + r-/cot/+/cose/. (i) THE SPIRAL. 37 If / 90, Bd = BA 4- Ac + ab + bd = .r f/ 4- /'-/(- cot /) +/cose / ; (2) Bd = -r f/ + T +/ cot / +/cose / ; (3) or in general calling, B,,, d, or Bd - t,; t, = Xf, + T T cot I + f cose I, (4) the + s ig n being used when / 90 and the - sign When / < 90. B, d t = B, A, + ^ /y c t + c, b. - d.g, - x i -f T +/ cose /-/cot /; (5) 7^ 90, A I ^ T I + COS / cose / + cot / - = : sin / but if 7 : 90, cos 7 = - cos 7 7-1 <. 7- T ~ COS I cose 7+ cot 7= . - . , sin 7 which we have seen = tan ]^1. Hence when f, = f, t, = t = X f/ +T+ f, tan %\ = x f/ +(! + f,)tanKI (") is a general equation whether 7 be greater or less than 90. Kx AMPLE 19. Given /. = 210 ; r = 818.8 ; f = 2.23 ; / -^= 40; OG, = 104.93 ; to find = Fig. (5): f />/ - '/ (/// /) tan ^7 - (104.93 821.03) .36397 = 403.75. With the same data TO FIND THE EXTERNAL SECANT. CASE i. When f 7 f, e.d, - (r // + f)secA / Dd,-r,,. (12) If f - 0, THE SPIRAL. CASE 2. e,, b, = P,, sec A,, D b, - P,, . CASE 3. f, = f , e/ d, - (P,, 39 (13) (14) PROBLEM II. TO FIND THE LOCATION OF THE OFFSET "f". % 22. Given two curves with radii r t and r llt a distance D t D tl d joining the points D t and D in also the angles BD,D tl = p and CD ll D, = e t to find the points A, and A lt at which a line drawn through the centers r t and r n at ^ and C will cut the curves A t D, and A H D tl . From Fig. 6 we have FC = h ^sinp-r y/ siu (180 - (/> -|- 0) ), (i) 4o TRANSITION CURVE. and since BC r t - (r lt 4-f t \ FC_ h _*/ ship- r,, sin (180- (/o+0)_ . BC~r,-(r tl +f) ~ r t -(r tt +f)~ D tl CD, = 2 = i8o-(p + 0), *-S = w =/?,,C .4,,; (3) whence r,,u = A lt D M (4) in which o> is an arc. If 2 > >1>, then the point ^4,, lies between D n and Z>, and the distance A tl D z is measured from D '., towards -#// to .4,,. If ^ > 2, the point ^f,, lies beyond D lt and the distance A n D tl is measured from D tl to ^ /; whence ^ /y is established. On a perpendicular to a tangent at A n and from A lt lay offy; and establish A t . When / is small, the direction of the radial line can be estimated near enough. The method of fixing B t and B in in Fig. 2, has already been indicated. If r t co, D t A, is tangent. Fig. 7 applies. EXAMPLE 20. r, = 1000 ; r n - 600 ; rf = 300 ; p = 70; .= 96; /-50': 1st to find A = rfsinp - ;- /y sin (180 - (p - 0) = 300 X .93969 - 600 y: .24192 = 136.75 = FC. k 136.75 2nd to find sm * = ^ _ ^ , y;) = -jjg^-.^-- = 3rd to find 180 [70 \- 96J = 14= 2; * 2 = w = 23 14= 9; w reduced to arc = -^-^77 = .1571. 57.29 ria = 600 X .1571 = 94.26 = A,, /),, , and since * : 2, -4 7/ /^ /y is measured from D,, towards Dj. If 2 = D^C D^ t then 2 i.- 1 * and A lt D^ would be measured from 7>3 towards D u to establish point ^4 y/ ; A, is on a perpendicular to a tangent to the curve with radius r,, at A,,. THE SPIRATv. 41 PROBLEM III. 3 I 2. To unit two circular curves turning in the same direction, whose centers are a fixed distance apart, by the introduction of a third circular curve of an intermediate radius and to unit the same with the main curves by the use of transition curves : Given two circular curves with radii r, and r.,, respectively, whose centers are apart a distance A C b = r i ~ ( r z +.///) i atl( i which are separated from each other a distance D t D n = f tl . It is desired to introduce between them a third curve with radius r lt , less than r, and greater than r.^ and to join the curve with radius r lt with those having radii r, and r.. by means of transition curves : By the figure AD, = AA, = AB t = r t = AC + CD + D lt D, = b + r, + f u , whence (i) 42 TRANSITION CURVE. AB = c = r,-(r,,+/,); (2) BC = a = r,,-(r 3 +/ 3 ); (3) a, b and c form the sides of a triangle ABC in which Any angle as ^4 may be found by the formulae, Ver A = 2 ~fc~ iu which s = -- (^+^+^ ; (6) Sin^ = -~-sin ^4; (7) Sin C = sin -4 ; (8) Reducing each of the above angles to arc as indicated in another part of this book we have, A, D, = r,A A. i D tl - r ;J 0; A lt A,, = r lt (iSo-B) (9) the arc - "" in the same way, THE SPIRAL. 43 (13) A,B, = r,ft; A tl B n = r,,ft,; A 3 B, = r,,j8 3 ; AiBt = r 8 flt! (H) J3 tl 3 =r it d; d = (iSo-) - (ft, + j8 3 ); (15) ,/>, - r, J. + r, ft - r, (4 + ft); (16) EXAMPLE 21. Given r ; =2865; r /y =1146; r 3 = 716.3; / // =20; />= 171900; A- 90; L 3 = 90 ../ / =/ 3 = .17 to join the circular curves ^Z> y and # 4 />> by Prob. III. First finding a, b and c by (1-3) we have by (6) ver A = 344' ; arc A = .06516; by (7) (180 ^)=1848'; arc (180 - )=.32811; by (8) C- 1506'; arc C=. 26054; ^/ A= 2865 >< .06516 = 186.68; AD,,= 716.3 X-26054 = 186.77; A,,A- A =1146X.3281= 376 ' 00; ^= - = .0157; ^4;^=^ lt B tl =A* 3 =A 4 = o.Q; 5=18 48'-(2 15'+2 150 =1420'; arc 6 =.25017; B n JS 3 = 1146X. 25017 =286.7; B, #,=2865 (.06516 f. 0157) =231.67; B D n =716.3 (.26054+ .06282) =231.62. If we make f, = o and r, = r tl , then c o, 6' = o, f tl and ^ It Q^b = a and is coincident with it, J. = 180 - B and the problem reduces to uniting A tl A^ with D lt B by means of the transition curve. If r n = r, , c = o, C = o, in which case f, and f 3 = o'a is equal to and coincides with b and J. 180 - It, equations for vers = o. The problem reduces to uniting curves B, D t and B D tl by means of a transition curve whose length L 2 = If f 3 and/ 7 = o while r, , r lt and r 3 retain their values, the transition curves disappear and the curve with radius r lt compounds at A 4 and A, with the curves having radii r, and r 3 . 180-^ is the central angle of A tl A 3 ; (18) C = that of D H A\ A = that of A, D, (19) *For determining large central angles of /3, , (3,, , /3 ;Jj and j8, t the method of 16 is to be preferred. 44 TRANSITION CURVE. PROBLEM IV. I 22. Given two curves turning in the same direc- tion, whose centers are a distance apart = D t D tl and whose radii are r, and r n . To fix the position of a tan- gent A, A in and connect it with the circular curves A n C, and AC lt by means of transition curves having a fixed Re- value of P (Fig. 9). Lst C, C tl be a line joining any two points C, and C n of the circular curves with radii r t and r n \ D, D,, the known distance between the centers of the curves r, and r n . Measure the angles D t C, C tt and C, C,, >. By traverse we find the angles C I D,D II and C^D^D,. If from D t and D tl we let fall perpen- diculars to any imaginary tangent passing through the origins of the transition curves, by the conditions of the problem the length of these perpendiculars will be: D. A, = r, +f, and D tt A ni = r tl +/ , and any distance D, K t = (r, +/) - ( r ,, + //), (i) THE SPIRAI,. 45 and if we denote the distance D, D tl by D 4 , then G 7 , Z>, ^// = difference of G r , Z>, D,, and 0. C t D, A tl reduced to arc and multiplied by r, = the distance C, A lt to establish A tl , from A fl lay off f t normal to the circle and establish A t . In the same way C lt D /t A = the difference between 180 and C tl D,, D, . GffD^A^ multiplied by r tl arc distance G,,A n from G lt to A 4 to establish A. it from which lay off f lt and establish A tll \ then a line through A, A in is the required tangent. The distance A t O, to the origin 0, = x, - r, sin a t and the distance A in O u = x n - r,, sin a n \ whence the distance 0, O n = D, D tl sin 6 - [(* - r tl sin a y/ )+(.r / - r, sin a ; )]. (3) If we wish the distance 00 = zero, make I) i D n \_(x n - r n sin a l ^-\-(.r l - r, sin a y )] cose ^. (4) This gives the shortest possible distance between the centers of the circular curves where the transition curves are introduced. The transition curves may have different values of P provided A, O t -\- A in O u or < than A,A IU . If the curves are reversed, D t K n = (r, +//) 4- (r tl +///) and we find the value of 9 by completing the traverse 0, C HI D tll D, C r EXPLANATION OF THE TABLES. BY RECTANGULAR CO-ORDINATES. $ 25. Tables I to IX give values for laying out the transition curve by the method of rectangular co-ordinates, They are equally applicable for uniting a tangent with a circular curve, or curves having different radii, by means of the transition curve. L lt and L, may be taken sepa- rately from the same column as also may a,, and a, , and their difference will be the value of L and for the length and central angle respectively. The several ordinates, JT, y t x f , y f , are laid off from B, or B n as origin (Fig. 2) with arc B,A t or B tl A lt as axis, the same as if B, or B n were written for A in Fig. 3, and the successive stations were B, C } D, etc. and B, , C t , D, , etc., successive points x, x, , x ni , etc. with corresponding y,y, , y tl , etc. , values normal to the curved axis A, B, A tl B lt in the same manner as if A, U, were tangent. Values of d are dropped, as in no case in these tables do they reach j\ y of a foot. Table II is extended as an example to develop these differences to an amount worthy of consideration. P and v are taken of such values as to avoid introducing fractions in L. L is supposed to be measured on the curve, but since the chords are generally quite short, the sum of their lengths is but little less than that of the curve, hence no allowance is made for the length of the curve being in excess of the sum of the lengths of the chords. EXPLANATION OF THE TABLES. 47 EXAMPLE I. Given I, - 120 ; r, - 1432.5; r n = 716.3 (Table II), then a t/ a, =s 9 36' 2 24' = 7 12' = < with E as origin. At the end of the first chord length from E towards / we have x = 30', y = .03 = co-ordinates for F; x, 60, y, = .21 = co-ordinates for G; Xll = 90, y,, = .70 = co-ordinates for H; a; 3 119.98, r 3 = 1.67 = co-ordinates for /; x t = 60; y f = .21; /= 42. The method of laying out these co-ordinates is shown in Fig. 3, in which the origin A corresponds to the point E in this example, and A, D, becomes a curved axis with a radius of 1.432.5. If the transition curve were laid off from B,, A n as axis and ,, as origin, the above values x &c and y &c would be just the same except they would be laid off from the convex side of A,, R n instead of from the concave side, as was the case with A, B, as axis. If the curvature of the circular curve is of fractional degree the value of /and the last values of x, y and a wilt have to be computed by the formulae at the head of the respective column in the table. TABLE la TO Va BY DEFLEXION. 26. Given a tangent at any point of a transition curve as D to locate any other points as A, B y C, E, F, G. As in the case of the Tables for rectangular co-ordinates they are equally applicable to locating the points of the transition curve uniting a tangent to the circular curve or circular curves of different radii with each other. (Fig. 4) EXAMPLE 2. I^et L,, L t =*L= 120 be the length of the transition curve; r/ = 1910; r n =955, the radii of the circular curves to be united by L by deflections from a tangent at D, where the curvature corresponds to r,. The tangent will be common to the circle and transition curve whose rate of curvature "Z>" (Table II) =-3. DE, EF, FG etc., being chords of the transition curve each 30 feet. The tangent at D is a tangent to the circular arc with r/ 1910. Then by the formula w = D A L -f- - , in which o /; - 3; A = .?- ; DE^L = ^ feet ; J = 3' ; 3 X.3' X 30 + 003' = 1U o w = 27' + 03' = 30' - the deflection from tangent at D to locate points. If we want to locate the point F from D, then L = 60 ; D = 3 A = 13' ; ' = 12' ; o> = 54 ^ 12 = 66' = 1 06' = deflection from 48 EXPLANATION OF THK TABLES. tangent at D to locate F with measurement from D to E thence E to F. If we wish to locate the points C, B and A from a tangent at D, then the deflection for any point C ' = the deflection for 30 feet for a 3 degree curve minus the deflection for the transition cnrve from A to , or o>=/?AL--=3X3X30- 4 =27'- 3'= 24'. o o SO' The deflection for the poiut ^ = 3 X-3'X 60 - -- = 054'-012'=042. o If we have run the curve from A to D and changed the instrument to D in order to place the line of sight tangent to D, take a back sight on A and deflect 54' we have a tangent at Z>. To facilitate the use of the tables it is best to set the vernier at 54' and set the telescope on line AD, turn the vernier to O and continue deflection as tabulated, reading downward from Z>, locating the points E, F, G and C. If the curve is being run from D towards A then set the vernier at the angle indicated for any angle G, when back sight is on G from D deflect from zero and continue to deflect the angles tabulated in succession, reading up the column from D to locate C, S and A. The degree of the curvature at the instrument point controls the deflections either way. The above explanation enables us to run the transition curve from the point of greatest radii to that of its least radius, and vice versa. To use the tables when the beginning or end of the transition curve merges into a circular curve of a frac- tional degree, as joining a 2^ degree curve with a 6 degree curve, or a 2 degree curve with a 5^ degree curve ; compute for 15 feet, then 2.5 X .3' X 15 + oo'.ys = o 12', which locates a point D at 3 degrees curvature ; then from D, with back sight on C -\- 15 ; 2.5 X .3' X 15 + oi'.so = o i2'.75 UNIVERSITY) EXPLANATION OF THE TKSffSS^' 49 will be the deflection to fix a tangent at D from which point proceed with deflections down the D column. If the last chord FG is one-half L, the deflections of the table can be followed up to F to which point the instru- ment can be moved and tangent established from which deflect w = DA + oo'.75 = 5 X .3' X 15 + oo'.75 = o 23'. If it is desired to deflect the whole angle from D to G when F G = one-half L when L f = the length from DtoF, Z G = /:, + -, then 2 r = 2296 ; -T-? X 57 18' = 5 = o 19'; *> = D A L f + - + - X 57 D 18' = 3 X .3' X 75 + 19' - 67' + 19' = i 26' . Any other fraction of L than I /L may be used in the formulae. BY RECTANGULAR CO-ORDINATES. 000000 OOOiCOOXCC O O i-' d 10 1C 1^ C: CO 'g gi? (N g ^ J ffl D < o / O O". -J r-l rl 1^ O M TP I> rl VC O O O O Oi ** sC co o i- a ?s m h co S o So o ' rH (N 1C O C 'O 1^- (>1 T 1 C >O CD CO ' rH CO ICJ CO Co' '- is^ rH 04 CO t^ C5 BY RECTANGULAR CO-ORDINATES. 000000 ooioooooeo 2 1 O iO O O - T 1 O o S ^r oc ^ 51 co j f xilS 0000 0000 OOOO O CO -f O O O i^ O Ci CS 7 |Tr o oo oo T CO CO O OJ : o w^Sl?? = 3 R ^ t 1 1 1 l o T-* r-( ; v ~ 2 00 t* Tfl T?< CO O CO - OO O (M C^ CM 1C OO II II f bb^cb ^ O b b r-l rH C^ CO rf i (^ a O I < ii > I 0^ || HiH O 'O O iO O LO O 1 CO Tfi CO t^ O5 O (M ; g 4J U-i O o^caco^ioco rH O (NlOCiuOrHCiCOCOO 9 O SI I I 1 > < I !l I 8888888S8 OOOt-^THrHC^CO^TjJlO^Ot^-oi ' b * I '5 j \ \ ! 88888888 M M M OiOO * ^ -( t-i (N CM !M 5 ^ O W J DQ O s i w^ 111 J ffl 8 cs OCOOOiOOO r, N< II i ^ II h H * ^ Q ^ ssftgssffe h -I O5 CO ^-< -f< u I iH rHMi * IH O^ I 89 8 2S8 rl T-H r-i r-H i-l C . 6*" I 54 BY RECTANGULAR CO-ORDINATES. s ^ OOOOCiOCCO O O iO O TtJ O rp CT. I- oi O -" Cr Q C> * BY 55 A = deflection angle per foot for 1 circular curve. 5 = -^deflection angle from tangent at A towards G for points o of spiral. D = degree or rate of curvature at position of instrument. o> = deflection from tangent at any point of spiral to any other point of spiral. Then o> = D A L + -4-= D A L -^-57 18' 3 6 r -f sign for running towards G sign towards A. The instrument point is the origin of/, and x. x conceived to to be laid off on the circular curve passing through instrument point as axis <0 * S 5 8 o ^a^j T* r,< oO CO CN ^ ^ CN 1C ^ -T - CC CM rH fj 5 c% 8 CM ^ S J..SN i: ^ "* -* CO CM ^ ^ t2 ^ b 36 b - ?- b CO CO CM CM ?1 tn ^ H b CM 06 00 CM Cfl 00 CM no CM rH 55 r-t TJI CM CM ^ ^ b j b ^ CM "^ A *O ^ 1 V. i* K ...... CO CM O 00 Cd CO CM ^ .S 5 ^ TH rH rH b b ^- CM CO . a 'O TJI CM b C^ ii 3 ^ O iH fH CM gc | ^ ^ fl'O ^H oo *- 1 o c CM T CM O ^ 1 3 8 b o g b b 2-i CM CM CO b ^M O -^ Tf O CM CO O CM O "!t s '- b 3 ^ b b b SH CM CM CO j i 3 CO II S^ 8 o b b b ?i ^H CM CO ii cb b OS CM ib 00 ^-1 rf a. * p p o p p o p P ^-( CM co ^ So b i- 00 < m u Q W fr O W ** * o o CO O O5 CM ci OJ O rl ^J t^ CO Nj So o CO ^ S CM CM BY DEFLECTIONS. 57 BY DKFtKCTIONS. & S CQ S ll *0 C3 Oi ^1 CO Cl OJ cb b o b O 000 rH iH O is & *' HJ b b ^ b b ?3 u o a JU J, 00 iH O rfi rf CO bob s 2^ a I' b b b rH is f S o to 11 fe || 3 b b b g ib cb ^ ^ rH CO . U3 bob ob 5 II 3 01 ^ rt b b S ^ 5 S b b b TH rH GO rH ll cb ca b iO I> C-l O rH C^ id QO rH rH rH pop p P ? <1 P3 U Q W fc O s So H " 8 S ^ V V , ? 1 O CJ - Tj< CD t~ O5 BY DEFLECTIONS. 59 CO CO- CO CM CM -H ^ S? O CM O - f P c ^ T* U ^ w * ^ r p o s s CM I 3 C5 CO 1-1 CO rHcSi I i iCO'TiJQT i CO O o b b b b b b TH 2-i ^ ! O CO C CT> CM i-O 00 r-J -T I> O o o o o r i >i -H CM" ~i ^i c? OOOOOOOOOOOi S g2 6o PROBLEM V. To introduce the transition curve in alignment where simple curves have been run. In Fig 10, suppose ABC, a simple circular curve, to have been run tangent to the line O G tl at A with radius r l = D l A=D,B; it is desired to introduce a spiral whose greatest curvature has a radius r n - D n A tl , D tl B n , D tt B = AB; r n = A tl B AB = the distance to measure from PC to locate B\ BA n the distance to measure from B on BA lt to locate A n . From m and r tl we have 24 m r lt , squaring, 4 r? a,/ 2 24 m r lt 6w . L 2 r tl a = L = l r M (0 /y - a /y ) = .5^, ; (r - The rate of curvature of r tt should not be more than from i to 2 greater than that of r, (when possible) for curves of a curvature less than 10; 2 to 3 difference, for 10 to 15 rate of curvature; 3 to 5 difference for 15 to 20 rate of curvature.