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 G1 Hermite interpolation method for quintic PHoPH curves was recently reported. This article addresses the C1 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 C1 PHoPH interpolators depend on the configuration of the C1 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 C1 PHoPH splines, which converges to given reference curves with the approximation order 4.