An approach to geometric interpolation by Pythagorean-hodograph curves

Abstract

The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree n is studied independently of the dimension d ≥ 2. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.

C1Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

“…Those results were later generalized to some level in [13]. The most general results on this type of interpolation can be found in [14,15]. The problem of C 1 and C 2 Hermite interpolation by spatial PH curves of degree ≥5 has been studied in [16][17][18][19].…”

“…However, low degree polynomial interpolating splines are important in several applications, and one would insist on using cubic curves. One possibility is to relax C 1 continuity to G 1 continuity but unfortunately it is not always possible to interpolate G 1 Hermite data by PH cubic (see [12,14,15], e.g.). To avoid this, C 1 Hermite interpolation by spatial PH biarcs will be considered in this paper.…”

Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

“…Nevertheless, the Pythagoreanhodograph (PH) curves have a distinct advantage over ordinary polynomial curves in this respect, since their arc lengths are simply polynomial functions of the curve parameter [9]. Many algorithms for the construction of planar and spatial PH curves have been developed, e.g., [13,17,24,25,27,30,35,37]. The intent of the present paper is to investigate the feasibility of constructing surface patches with Pythagorean-hodograph isoparametric curves.…”

Identification and “reverse engineering” of Pythagorean-hodograph curves

“…Many methods for the construction of planar and spatial PH curves are available [8,12,14,20,21,22,23,24,25,27,29,30]. The goal of this study is to facilitate their importation into commercial CAD systems through existing CAD data formats, by developing algorithms that (i) identify whether or not specified Bézier/B-spline data define a PH curve; and (ii) if so, reconstruct its "internal structure" variables.…”