Construction and shape analysis of PH quintic Hermite interpolants

Abstract

In general, the problem of interpolating given first-order Hermite data (end points and derivatives) by quintic Pythagorean-hodograph (PH) curves has four distinct formal solutions. Ordinarily, only one of these interpolants is of acceptable shape. Previous interpolation algorithms have relied on explicitly constructing all four solutions, and invoking a suitable measure of shape—e.g., the absolute rotation index or elastic bending energy—to select the “good” interpolant. We introduce here a new means to differentiate among the solutions, namely, the winding number of the closed loop formed by a union of the hodographs of the PH quintic and of the unique “ordinary” cubic interpolant. We also show that, for “reasonable” Hermite data, the good PH quintic can be directly constructed with certainty, obviating the need to compute and compare all four solutions. Finally, we present an algorithm based on the subdivision, degree elevation, and convex hull properties of the Bernstein form, that gives rapidly convergent curvature bounds for PH curves, using only rational arithmetic operations on their coefficients.