Construction of C 2 Pythagorean-hodograph interpolating splines by the homotopy method

Abstract

The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing a C 2 PH quintic “spline” that interpolates a given sequence of points p 0 , p 1 ,..., p N and end-derivatives d 0 and d N to be reduced to solving a “tridiagonal” system of N quadratic equations in N complex unknowns. The system can also be easily modified to incorporate PH-spline end conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2 N +1 distinct interpolants for each of several different data sets. We observe empirically that all but one of these interpolants exhibits undesirable “looping” behavior (which may be quantified in terms of the elastic bending energy , i.e., the integral of the square of the curvature with respect to arc length). The remaining “good” interpolant, however, is invariably a fairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding “ordinary” C 2 cubic spline. Moreover, the PH spline has the advantage that its offsets are rational curves and its arc length is a polynomial function of the curve parameter.