Hermite interpolation by Pythagorean hodograph curves of degree seven

Abstract:

Polynomial Pythagorean hodograph (PH) curves form a remarkable subclass of polynomial parametric curves; they are distinguished by having a polynomial arc length function and rational offsets (parallel curves). Many related references can be found in the article by Farouki and Neff on $C^1$ Hermite interpolation with PH quintics. We extend the $C^1$ Hermite interpolation scheme by taking additional curvature information at the segment boundaries into account. As a result we obtain a new construction of curvature continuous polynomial PH spline curves. We discuss Hermite interpolation of $G^2[C^1]$ boundary data (points, first derivatives, and curvatures) with PH curves of degree 7. It is shown that up to eight possible solutions can be found by computing the roots of two quartic polynomials. With the help of the canonical Taylor expansion of planar curves, we analyze the existence and shape of the solutions. More precisely, for Hermite data which are taken from an analytical curve, we study the behaviour of the solutions for decreasing stepsize $\Delta$. It is shown that a regular solution is guaranteed to exist for sufficiently small stepsize $\Delta$, provided that certain technical assumptions are satisfied. Moreover, this solution matches the shape of the original curve; the approximation order is 6. As a consequence, any given curve, which is assumed to be $G^2$ (curvature continuous) and to consist of analytical segments can approximately be converted into polynomial PH form. The latter assumption is automatically satisfied by the standard curve representations of Computer Aided Geometric Design, such as Bézier or B-spline curves. The conversion procedure acts locally, without any need for solving a global system of equations. It produces $G^2$ polynomial PH spline curves of degree 7.