Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces

As observed by Farouki et al. [9], any set of C space boundary data (two points with associated first derivatives) can be interpolated by a Pythagorean hodograph (PH) curve of degree 5. In general there exists a two dimensional family of interpolants. In this paper we study the properties of this family in more detail. We introduce a geometrically invariant parameterization of the family of interpolants. This parameterization is used to identify a particular solution, which has the following properties. Firstly, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Secondly, it has the best possible approximation order (4). Thirdly, it is symmetric in the sense that the interpolant of the “reversed” set of boundary data is simply the “reversed” original interpolant. These observations lead to a fast and precise algorithm for converting any (possibly piecewise) analytical curve into a piecewise PH curve of degree 5 which is globally C. Finally we exploit the rational frames associated with any space PH curve (Euler-Rodrigues frame) in order to obtain a simple rational approximation of pipe surfaces with a piecewise analytical spine curve and we analyze its approximation order.