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

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 , the new representation corresponds to a sequence of scalings/reflections of . The two representations are shown to be equivalent for cubic and quintic PH curves, when freedom in choosing 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 . In the context of the spatial PH quintic Hermite interpolation problem, variation of the unit vector 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.