A control polygon scheme for design of planar C2 PH quintic spline curves

A scheme to specify planar C2 Pythagorean-hodograph (PH) quintic spline curves by control polygons is proposed, in which the “ordinary” C2 cubic B-spline curve serves as a reference for the shape of the PH spline. The method facilitates intuitive and efficient constructions of open and closed PH spline curves, that typically agree closely with the corresponding cubic B-spline curves. The C2 PH quintic spline curve associated with a given control polygon and knot sequence is defined to be the “good” interpolant to the nodal points of the C2 cubic spline curve with the same B-spline control points, knot sequence, and end conditions—it may be computed to machine precision by just a few Newton–Raphson iterations from a close starting approximation. The relation between the PH spline and its control polygon is invariant under similarity transformations. Multiple knots may be inserted to reduce the order of continuity to C1 or C0 at specified points, and by means of double knots the PH splines offer a linear precision and local shape modification capability. Although the non-linear nature of PH splines precludes proofs for certain features of cubic B-splines, such as convex-hull confinement and the variation-diminishing property, this is of little practical significance in view of the close agreement of the two curves in most cases (in fact, the PH spline typically exhibits a somewhat better curvature distribution).