Point-Clothoid Distance and Projection Computation

A comprehensive algorithm to compute the minimum distance between a given point in the plane and an assigned clothoid spiral curve is herein proposed. The projection of the point on the curve is also solved. The solution is relevant in many applications ranging from robotics to autonomous vehicles. The minimization is formulated as a root-finding problem which typically has multiple solutions associated to local minima. A proper interval for the curvilinear abscissa, that contains the global solution is recognized. Due to its spiraling path, the clothoid has a low curvature region near the inflection point and a high curvature region around points at infinity, where the revolving curve shows many potential solutions. The transition from a clothoid to an arc of circle and from an arc of circle to a straight line---which are particular cases of clothoids---is smoothly computed. The present algorithm is validated with extensive numerical tests and is proved much better than brute force algorithms. The present results confirm the better efficiency of the proposed method in terms of accuracy, convergence and computational times.