Construction ofC ² Pythagorean-hodograph interpolating splines by the homotopy method

Abstract

The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing aC ² PH quintic “spline” that interpolates a given sequence of pointsp ₀,p ₁,...,p _N and end-derivativesd ₀ andd _N to be reduced to solving a “tridiagonal” system ofN quadratic equations inN complex unknowns. The system can also be easily modified to incorporate PH-splineend 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 theelastic bending energy, i.e., the integral of the square of the curvature with respect to arc length). The remaining “good” interpolant, however, is invariably afairer 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 arerational curves and its arc length is apolynomial function of the curve parameter.