In laying out the curves on a railroad it will not do, on account of the high speed of trains, to pass abruptly from a straight stretch of track to a circular curve. In order to make the change of direction gradual, engineers make use of transition curves to connect the straight part of a track with a circular curve. Arcs of cubical parabolas are generally employed as transition curves.
Illustrative Example 3. The transition curve on a railway track has the shape of an arc of the cubical parabola . At what rate is a car on this track changing its direction (1 mi. = unit of length) when it is passing through (a) the point (3, 9)? (b) the point ? (c) the point ?
103. Radius of curvature. By analogy with the circle (see (38), § 99), the radius of curvature of a curve at a point is defined as the reciprocal of the curvature of the curve at that point. Denoting the radius of curvature by R, we have
↑Hence the radius of curvature will have the same sign as the curvature, that is, + or -, according as the curve is concave upwards or concave downwards.
↑In § 98, p. 152, (43) is derived from (42) by transforming from rectangular to polar coordinates.
![{\displaystyle R={\frac {\left[1+\left({\frac {dy}{dx}}\right)^{2}\right]^{\frac {3}{2}}}{\frac {d^{2}y}{dx^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1fb3ec5cba15681132dc68bab5dc69b31e5b8b2)
![{\displaystyle R={\frac {\left[\rho ^{2}+\left({\frac {d\rho }{d\theta }}\right)^{2}\right]^{\frac {3}{2}}}{\rho ^{2}-\rho {\frac {d^{2}\rho }{d\theta ^{2}}}+2\left({\frac {d\rho }{d\theta }}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8883d80af28d09b513a0712a21520aeec93356)
