Regular curves and proper parametrizations

We present an algorithm for determining whether a given rational parametric curve, defined as vector valued function over a finite domain, has a regular parametrization. A curve has a regular parametrization if it has no cusps in its defining interval. It has been known that the vanishing of the derivative vector is a necessary condition for the existence of cusps. We show that if a curve is properly parametrized, then the vanishing of the derivative vector is a necessary and sufficient condition for the existence of cusps. If a curve has no cusps in its defining interval, its proper parametrization is a regular parametrization. We present a simple algorithm to compute the proper parametrization of a polynomial parametric curve which is used to analyze for cusps and later on reduce the problem of detecting cusps in a rational curve to that of a polynomial curve.