Rational curves over generalized complex numbers

Abstract

Complex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with respect to Möbius transformations, special implicit forms, an easy detection procedure, and a fast algorithm for computing their μ-bases. But only certain very special rational planar curves are also complex rational curves. To construct a wider collection of curves with similar appealing properties, we generalize complex rational curves to hyperbolic and parabolic rational curves by invoking the hyperbolic and parabolic numbers. We show that the special properties of complex rational curves extend to these hyperbolic and parabolic rational curves. We also provide examples to flesh out the theory.