$C^2$ Hermite interpolation by Pythagorean Hodograph space curves

Abstract

We solve the problem of C 2 Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of C 2 space boundary data (two points with associated first and second derivatives) we construct a four-dimensional family of PH interpolants of degree 9 and introduce a geometrically invariant parameterization of this family. This parameterization is used to identify a particular solution, which has the following properties. First, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Second, it has the best possible approximation order 6. Third, it is symmetric in the sense that the interpolant of the reversed set of boundary data is simply the reversed original interpolant. This particular PH interpolant is exploited for designing algorithms for converting (possibly piecewise) analytical curves into a piecewise PH curve of degree 9 which is globally C 2 , and for simple rational approximation of pipe surfaces with a piecewise analytical spine curve. The algorithms are presented along with an analysis of their error and approximation order.