Clifford algebra, spin representation, and rational parameterization of curves and surfaces ∗

Abstract

The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts gives rise to different combinations of polynomials that satisfy further complicated identities. We present a novel approach to the Pythagorean hodograph curves, based on Clifford algebra methods, that unifies all known incarnations of PH curves into a single coherent framework. Furthermore, we discuss certain differential or algebraic geometric perspectives that arise from this new approach. In this paper, we present an approach that unifies all known incarnations of the socalled Pythagorean hodograph curves into a single coherent framework, through the use of Clifford algebra. As we shall see, Pythagorean hodograph (PH) curves are characterized by a certain Pythagorean n-tuple identity in the polynomial ring, relating derivatives of the curve coordinate functions. Thus far, the PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts involves rather different combinations of certain sets of polynomials. Currently, the diverse algebraic forms for PH curves in spaces with different dimensions and metrics are something of an enigma, that suggest the presence of a deeper underlying structure. The approach expounded in this paper reveals that all the different forms of PH curves can be expressed via a certain map, which we shall call the PH representation