Control point modifications that preserve the Pythagorean–hodograph nature of planar quintic curves

Abstract

Although planar Pythagorean–hodograph (PH) curves are compatible with the standard Bernstein–Bézier representations, freely modifying the control points will compromise their PH nature. The present study focuses on identifying control point displacements that ensure a given planar PH curve remains a PH curve. In particular, for planar quintic PH curves $r\left(t\right)$, $t\in [0,1]$ it is shown that finitely-many simultaneous displacements of two control points yield modified quintic PH curves, identified as the solutions of quadratic and cubic equations. As a more practical approach, modification of PH quintics in canonical form with $r\left(0\right)=0$ and $r\left(1\right)=1$ by the displacement of a single interior control point is considered, with the remaining interior control points being used to minimize a measure of deviation from the original PH quintic. As illustrated by several examples, this approach provides an efficient and intuitive means of effecting reasonable shape modifications within the space of planar quintic PH curves.