Spatial quintic Pythagorean-hodograph interpolants to first-order Hermite data and Frenet frames

Abstract

We present a scheme to find spatial quintic Pythagorean-hodograph (PH) curves that interpolate given first-order Hermite data and Frenet frames. The two free parameters that appear in general quintic PH interpolants are determined to adjust the orientation of binormal vectors. Using the unique cubic interpolant to the given set of Hermite data as a reference, we produce a quintic PH interpolant that shares not only the first-order Hermite data but also the Frenet frames at the endpoints with the cubic counterpart. This approach can be readily applied to a sequence of points to generate a quintic PH C1 spline curve with Frenet-frame continuity. We also prove that for any nonlinear analytic curve, such PH Frenet interpolants uniquely exist if the Hermite data are extracted from sufficiently close points on the curve.