Identification of two classes of planar septic Pythagorean hodograph curves

Abstract

Polynomial Pythagorean hodograph (PH) curves have a polynomial arc length function and rational offsets, which distinguish PH curves from general polynomial parametric curves. However, these algebraic properties can hardly be used directly for the identification of PH curves. In order to determine whether or not a given septic planar polynomial curve is a PH curve, Zheng et al. (2016) studied a class of septic curves’ geometric properties and proposed an efficient algorithm. In this paper, we further complete their work on Bézier control polygons of septic PH curves. We point out that there are three classes of septic PH curves according to different factorizations of their derivatives. Except the first class which has been studied, geometric characteristics of the other two classes are proposed. By introducing auxiliary points, the results are in terms of angles and lengths of legs of their Bézier control polygons. Moreover, we give feasible methods for the construction of auxiliary points, including various degenerate cases.