Interpolation of planar G1 data by Pythagorean-hodograph cubic biarcs with prescribed arc lengths

Abstract

The interpolation of two points and two tangent directions by planar parametric cubic curves with prescribed arc lengths is considered. It is well known that this problem is highly nonlinear if standard cubic curves are used. However, if Pythagorean-hodograph (PH) curves are considered, the problem simplifies due to their distinguished property that the arc length is a polynomial function of its coefficients. Since a single segment of a PH cubic curve does not provide enough free parameters, the so called PH cubic biarcs are used. A detailed and thorough analysis of the resulting system of nonlinear equations is provided and closed form solutions are given for any possible configuration of given data. The lookup table of the solutions is constructed enabling an easy implementation of the described method. Some quantities arising from geometric properties of the resulting curves are suggested in order to select the most appropriate one and the bending energy is confirmed as the most promising selection criterion. Several numerical examples are presented which confirm theoretical results. Finally, an example of approximation of an analytic curve by G1 PH cubic biarc spline curve is presented and the approximation order is numerically established.