G1 fitting with clothoids

A new algorithm for the solution to the problem of Hermite G1 interpolation with a clothoid curve is herein proposed, that is, a clothoid that interpolates two given points in a plane with assigned unit tangent vectors. The interpolation problem is formulated as a system of three nonlinear equations with multiple solutions, which is difficult to solve even numerically. In this work the solution of this system is reduced to the computation of the zeros of only one single nonlinear function in one variable. The location of the relevant zero is tackled analytically: it is provided the interval containing the zero where the solution is proved to exist and to be unique. A simple guess function allows to find that zero with very few iterations in all of the possible instances. Computing clothoid curves calls for evaluating Fresnel-related integrals, asymptotic expansions near critical values are herein conceived to avoid loss of precision. This is particularly important when the solution of the interpolation problem is close to a straight line or an arc of circle. The present algorithm is shown to be simple and compact. The comparison with literature algorithms proves that the present algorithm converges more quickly and accuracy is conserved in all of the possible instances, whereas other algorithms have a loss of accuracy near the transition zones.