Detecting cusps and inflection points in curves

Abstract

In many applications it is desirable to analyze parametric curves for undesirable features like cusps and inflection points. Previously known algorithms to analyze such features are limited to cubics and in many cases are for planar curves only. We present a general purpose method to detect cusps in polynomial or rational space curves of arbitrary degree. If a curve has no cusp in its defining interval, it has a regular parametrization and our algorithm computes that. In particular, we show that if a curve has a proper parametrization then the necessary and sufficient condition for the existence of cusps is given by the vanishing of the first derivative vector. We present a simple algorithm to compute the proper parametrization of a polynomial curve and reduce the problem of detecting cusps in a rational curve to that of a polynomial curve. Finally, we use the regular parametrizations to analyze for inflection points.