Design of C 2 spatial pythagorean-hodograph quintic spline curves by control polygons

TLDR: An intuitive approach to designing spatial C2 Pythagorean---hodograph (PH) quintic spline curves, based on given control polygons, is presented.

Abstract: An intuitive approach to designing spatial C2 Pythagorean---hodograph (PH) quintic spline curves, based on given control polygons, is presented. Although PH curves can always be represented in Bezier or B---spline form, changes to their control polygons will usually compromise their PH nature. To circumvent this problem, an approach similar to that developed in [13] for the planar case is adopted. Namely, the "ordinary" C2 cubic B---spline curve determined by the given control polygon is first computed, and the C2 PH spline associated with that control polygon is defined so as to interpolate the nodal points of the cubic B---spline, with analogous end conditions. The construction of spatial PH spline curves is more challenging than the planar case, because of the residual degrees of freedom it entails. Two strategies for fixing these free parameters are presented, based on optimizing shape measures for the PH spline curves.