Construction and shape analysis of PH quintic Hermite interpolants

In general, the problem of interpolating given first-order Hermite data ten d 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 ex plicitly constructing all four solutions, and invoking a suitable measure o f shape-e.g., the absolute rotation index or elastic bending energy-to sele ct the "good" interpolant, We introduce here a new means to differentiate a mong 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" c ubic interpolant. We also show that, for "reasonable" Hermite data, the goo d PH quintic can be directly constructed with certainty, obviating the need to compute and compare all four solutions. Finally, we present an algorith m 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. (C ) 2001 Elsevier Science B.V. All rights reserved.