Spatial Pythagorean Hodograph Quintics and the Approximation of Pipe Surfaces

Abstract

Abstract. As observed by Farouki et al. [9], any set of C 1 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 1 . 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.

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

“…The ratio of the improvement is closer and closer to 64. As we have observed in [23], in the case of C 1 interpolation, the ratio of the improvement converges to 16. Clearly, instead of the simple uniform subdivision, using an adaptive subdivision scheme would reduce the number of segments.…”

“…a global method for C 2 interpolation of point data by quintic splines was presented in [13]. Hermite interpolation of G 1 boundary data was addressed in [17], and C 1 Hermite interpolation by PH quintics was discussed in [10,23]. In the latter case, there exists a family of interpolants to any C 1 Hermite data which depends on two free parameters.…”

“…As in the case of the C 1 Hermite interpolation [10,23], one of the angular parameters, say θ 2 , can be chosen to be zero due to the non-trivial fibers of the mapping preimage → hodograph (2.19). This fact is formally proved in the following lemma.…”

C 1 Hermite interpolation by Pythagorean hodograph quintics in Minkowski space

“…PH curves and their applications have been thoroughly investigated, cf. [8,10,16] and the survey [7].…”

“…The problem of C 1 Hermite interpolation with PH curves in R 3 yields a twoparameter family of PH quintics [8,16]. There exists a particular interpolant which is geometrically invariant, preserves symmetry and planarity, and possesses approximation order 4.…”