Efficient Solution of the Complex Quadratic Tridiagonal System for C2 PH Quintic Splines

Abstract

The construction of C2 Pythagorean-hodograph (PH) quintic spline curves that interpolate a sequence of points p0,...,pN and satisfy prescribed end conditions incurs a “tridiagonal” system of N quadratic equations in N complex unknowns. Albrecht and Farouki [1] invoke the homotopy method to compute all 2N+k solutions to this system, among which there is a unique “good” PH spline that is free of undesired loops and extreme curvature variations (k∈{−1,0,+1} depends on the adopted end conditions). However, the homotopy method becomes prohibitively expensive when N≳10, and efficient methods to construct the “good” spline only are desirable. The use of iterative solution methods is described herein, with starting approximations derived from “ordinary” C2 cubic splines. The system Jacobian satisfies a global Lipschitz condition in CN, yielding a simple closed-form expression of the Kantorovich condition for convergence of Newton–Raphson iterations, that can be evaluated with O(N2) cost. These methods are also generalized to the case of non-uniform knots.