C2 Hermite interpolation by Pythagorean Hodograph space curves

We solve the problem of C2 Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of C2 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 C2, 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.