C1 Hermite interpolation method for septic PHoPH curves

Abstract

Pythagorean hodograph(PH) curves, which are polynomial parametric curves with the polynomial speed functions, have been formulated and analyzed both on a plane and in a space separately. If a single curve satisfies both planar PH and spatial PH condition simultaneously, it is a spatial PH curve with the planar projection. This type of curves are called as PH over PH curves, or PHoPH curves, and a $G^1$ Hermite interpolation method for quintic PHoPH curves was recently reported. This article addresses the $C^1$ Hermite interpolation problem using septic PHoPH curve. Since the hodograph of a PHoPH curve is obtained by applying two successive squaring maps to a quaternion generator polynomial, the PHoPH curve is of degree $4n+1$ when $n$ is the degree of the generator. So the hodograph of a septic PHoPH curve is constructed not directly from a generator but from a generator and a quadratic common factor. After fixing most parameters in the quaternion generator using the end tangent data, we can streamline the problem into a system of nonlinear equations with three unknown variables, which can be readily solved by numerical methods. The existence and the number of $C^1$ PHoPH interpolators depend on the configuration of the $C^1$ Hermite data. We provide the results of extensive Monte-Carlo simulations for the feasibility analysis of this problem. We also present a few examples of $C^1$ PHoPH splines, which converges to given reference curves with the approximation order 4.