G1 interpolation of v-asymmetric data with arc-length constraints by Pythagorean-hodograph cubic splines

Abstract

The interpolation problem of a sequence of points with prescribed segment arc-length by PH cubic spline is addressed. It turns out to be a v-asymmetric G1 Hermite interpolation problem with specified arc length. Using rectifying control polygon and reduction to a canonical form, it is shown that the problem can be expressed in terms of finding the real solutions to a system of nonlinear equations in two variables. A detailed and thorough analysis of the resulting system of nonlinear equations and closed form solutions are provided for any given data. It is confirmed that this construction of G1 Hermite interpolants of specified arc length admits at least two formal solutions, both of which have attractive shape properties and some (if any) must be discarded due to undesired looping behavior. The algorithm developed herein offers simple and efficient closed-form solutions to a fundamental constructive geometry problem that avoids the need for iterative numerical methods. Moreover, the algorithm is illustrated by several numerical examples.