Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C2 Pythagorean– hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C2 quintic bases on uniform triple knots are constructed for both open and closed C2 curves, and are used to derive simple explicit formulae for the B–spline control points of C2 PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the B ́ezier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C2 to C1. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.