Low-Rank Rational Approximation of Natural Trochoid Parameterizations

Abstract

Arc-length or natural parametrization of curves traverses the shape with unit speed, enabling uniform sampling and straightforward manipulation of functions defined on the geometry. However, Farouki and Sakkalis proved that it is impossible to parametrize a plane or space curve as a rational polynomial of its arc-length, except for the straight line. Nonetheless, it is possible to obtain approximate natural parameterizations that are exact up to any epsilon. If the given family of curves possesses a small number of scalar degrees of freedom, this results in simple approximation formulae applicable in high-performance scenarios. To demonstrate this, we consider the problem of finding the natural parametrization of ellipses and cycloids. This requires the inversion of elliptic integrals of the second kind. To this end, we formulate a two-dimensional approximation problem based on machine-epsilon exact Chebhysev proxies for the exact solutions. We also derive approximate low-rank and low-degree rational natural parametrizations via singular value decomposition. The resulting formulas have minimal memory and computational footprint, making them ideal for computer graphics applications.