Transition curve: definition and need

A transition curve is a curve of gradually changing radius, provided between a straight and a circular curve (or between two circular curves), so that the radius changes from infinity at the tangent to $R$ at the circular curve.

Objectives and need

1. To introduce the centrifugal force gradually from zero to its full value, avoiding a sudden jerk to the passengers and the load.
2. To allow gradual introduction of the superelevation over the length of the curve.
3. To allow gradual introduction of the extra widening.
4. To improve the appearance of the road by avoiding a visible kink at the junction of the straight and the curve.
5. To reduce the chance of skidding, overturning and lateral shift of the vehicle, so that the speed is more uniform.
6. To keep the vehicle within its lane (a vehicle naturally follows a transition path while entering a curve).

Types of transition curve

1. Spiral (clothoid / Euler spiral) - the radius varies inversely with the length; recommended by IRC.
2. Cubic parabola - $y = x^3 /(6RL_s)$; practically the same as the spiral for small deflection angles, and widely used on railways.
3. Bernoulli's lemniscate - used on roads where the deflection angle is large (e.g. loops); its radius decreases more slowly than that of the spiral.