COMPOUND CURVES. 89. A compound curve, being merely a series of two or more simple curves, the manner in which it is located is by setting out its components separately, each P.C.C. (Point of Compound Curvature) being treated as a P.C. or P. T., the direction of the tangent at each P.C.C. being given by its Index-reading. As regards the notes, instead of keeping them for each curve independently, it is better to carry the Index-reading through continuously from the P.C. to the P.T., so that the reading for the P.T. equals half the total intersection-angle. The length and intersection-angle of each component curve should be entered in the notes, and also the total length and total intersection-angle. 90. To locate a compound curve when the P.C.C. is inaccessible. Suppose, as in Fig. 41, p (the P.C. C.) is inaccessible. The points e and d, if accessible, may then be found by inserting the value of the intersection-angle, in the case of each curve separately, in Equation 9, and thus obtaining for T the distances ad and be. Then from the tangent de the curve can be located by offsets, as already shown. If the points d and e are also inaccessible, select in the curve some convenient point f, and from it set off the offset ƒh= of vers fop (by Equation 20). Similarly, from a point in the other branch of the curve lay off an offset ik = qi vers iqp. We can then find the position of p by Equation 21; thus: 91. Given a simple curve ending in a tangent, to connect it with a parallel tangent by means of another and ac the required curve: then since C, the central angle, is the same for both curves, the above equation holds good also in this case. 92. To connect a curve with a tangent by means of another curve of given radius, 1. Let ac in Fig. 43 be the given curve which it is required to connect with a given tangent at b. Find the point a on the given curve which has its tangent parallel to the given tangent, and measure e: then, since 2. But if the radius of the required curve is less than that of the other curve, then, as in Fig. 44, find the point d_at the intersection of the tangent at b with the given curve ac, and observe the angle of intersection at d = aod; then 3. An analogous case is that shown in Fig. 45, where it is required to connect the curve ac with a tangent on the convex side by means of the curve pb. Then, as before, find d and observe the angle of intersection at d = aod; then Suppose in case 3 the point d were found to coincide with a; then we merely have the case of a Y located on the tangent db, in which case the above formula becomes 93. Given a compound curve ending in a tangent, to change the P.C.C. so that the terminal curve may end in a given parallel tangent without changing its radius. 1. In Fig. 46 let the radius of the terminal curve pb be greater than the radius of the other curve pa; then, A. If we want to shift the curve inwards to b', then to find p', the new position of the P.C.C., we have R FIG. 46. a B. If apb' were the given curve, and it were required to shift it outwards to b, then e cos o = cos o' and since in both cases pqp' = = 0 -0', we can thus find the position of p or p', as the case may be. 2. Suppose, however, the radius of the terminal curve bp is less than the radius of the other curve pa as in Fig. 46, and that it is required to shift the tangent (A) inwards to b: then b compound curve, and it were required to shift it outwards, Then since in both cases (A) and (B) pqp' = o' — o, we can find the position of p or p' as the case may be. 94. To connect two curves, already located, by means of another curve of given radius, As in Fig. 48, let R be the radius of the easier curve, and r the radius of the sharper curve. Find the tangent ab as shown in Sec. 83, and also the distance ab by direct measurement or calculation; then = op qs = ab cosec (aqs). = Then, since oq R and os op' -r, where op and op' are each equal to the radius of the required curve, we have the three sides of the triangle oqs, from which we can find the angle oqs (see Sec. 231); and Thus we can find the position of p. Similarly, we can find the position of p'; or we can calculate the angle at o, which does equally well. The radius of the required curve must exceed qs+R+r If R=r, then ab sin (aqp) = 2(op-R)* |
