Numerical Computation of Egg and Double-Egg Curves with Clothoids

Abstract

The clothoid, also known as Cornu spiral or Euler spiral, is a curve widely used as a transition curve when designing the layout of railway tracks and roads because of a key feature: its curvature is proportional to its length. The classical method to compute a clothoid is based on the use of Taylor expansions of sine and cosine functions, usually starting with zero curvature at the initial point. In this paper the clothoid is presented as the only curve with a constant rate of change of curvature, which parametrization can be obtained by solving an initial value problem. In this initial value problem the curvature at the starting point can be chosen, being able to develop simple, efficient, and accurate algorithms to connect two oriented circumferences by means of clothoids. These algorithms are presented as a useful tool for designing egg and double-egg curves in highway connections and interchanges