A geometric product formulation for spatial Pythagorean hodograph curves with applications to Hermite interpolation

Abstract

A novel formulation for spatial Pythagorean hodograph (PH) curves, based on the geometric product of vectors from Clifford algebra, is proposed. Compared to the established quaternion representation, in which a hodograph is generated by a continuous sequence of scalings/rotations of a fixed unit vector n ˆ , the new representation corresponds to a sequence of scalings/reflections of n ˆ . The two representations are shown to be equivalent for cubic and quintic PH curves, when freedom in choosing n ˆ is retained for the vector formulation. The latter also subsumes the original (sufficient) characterization of spatial Pythagorean hodographs, proposed by Farouki and Sakkalis, as a particular choice for n ˆ . In the context of the spatial PH quintic Hermite interpolation problem, variation of the unit vector n ˆ offers a geometrically more-intuitive means to explore the two-parameter space of solutions than the two free angular variables that arise in the quaternion formulation. This space is seen to have a decomposition into a product of two one-parameter spaces, in which one parameter determines the arc length and the other can be used to vary the curve shape at fixed arc length.