Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

Abstract

It is shown that, depending upon the orientation of the end tangents $\t_0, \t_1$ relative to the end point displacement vector $\Delta\p=\p_1-\p_0$, the problem of $G^1$ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points $\p_0,\ldots,\p_n$ in $\mathbb{R}^3$, compatible with a $G^1$ piecewise--PH--cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape--preservation properties of the resulting curves, is illustrated by a selection of computed examples.