Dynamic Evaluation of Exponential Polynomial Curves and Surfaces via Basis Transformation

Abstract

It is shown in "SIAM J. Sci. Comput. 39 (2017):B424-B441" that free-form curves used in computer aided geometric design can usually be represented as the solutions of linear differential systems and points and derivatives on the curves can be evaluated dynamically by solving the differential systems numerically. In this paper we present an even more robust and efficient algorithm for dynamic evaluation of exponential polynomial curves and surfaces. Based on properties that spaces spanned by general exponential polynomials are translation invariant and polynomial spaces are invariant with respect to a linear transformation of the parameter, the transformation matrices between bases with or without translated or linearly transformed parameters are explicitly computed. Points on curves or surfaces with equal or changing parameter steps can then be evaluated dynamically from a start point using a pre-computed matrix. Like former dynamic evaluation algorithms, the newly proposed approach needs only arithmetic operations for evaluating exponential polynomial curves and surfaces. Unlike conventional numerical methods that solve a linear differential system, the new method can give robust and accurate evaluation results for any chosen parameter steps. Basis transformation technique also enables dynamic evaluation of polynomial curves with changing parameter steps using a constant matrix, which reduces time costs significantly than computing each point individually by classical algorithms.

Construction and Evaluation of Pythagorean Hodograph Curves in Exponential-Polynomial Spaces

“…Thus, one of the main goals of this work is also to suggest a stable formulation of the normalized B-basis of the two exponential-polynomial spaces (or, more precisely, algebraic-hyperbolic spaces) that are most frequently encountered when working with nonpolynomial PH curves, so that numerical instabilities are avoided. Furthermore, for such spaces, we aim at proposing a novel evaluation algorithm that is stable for a wide range of the exponential shape parameter, in contrast to the dynamic evaluation procedure in [15], and has a lower computational time (see section 6), compared with the de Casteljau-like B-algorithm [1,7,8,9] (analogue of the de Casteljau algorithm for classical polynomial Bézier curves), and with the algorithm introduced by Woźny and Chudy in [13].…”

“…A note on a fourth algorithm. We conclude this section with a short discussion about the dynamic evaluation algorithm presented in [14,15] which, although it can be specialized for EPH curves, presents stability issues for large values of ω. To explain why this is the case, we begin with a brief review of the method.…”

Construction and evaluation of PH curves in exponential-polynomial spaces

“…A note on a fourth algorithm. We conclude this section with a short discussion about the dynamic evaluation algorithm presented in [14,15] which, although can be specialized for EPH curves, presents stability issues for large values of ω. To explain why this is the case, we begin with a brief review of the method.…”

“…Thus, one of the main goals of this work is also to suggest a stable formulation of the normalized B-basis of the two exponential-polynomial spaces (or, more precisely, algebraic-hyperbolic spaces) that are most frequently encountered when working with non-polynomial PH curves, so that numerical instabilities are avoided. Furthermore, for such spaces, we aim at proposing a novel evaluation algorithm that is stable for a wide range of the exponential shape parameter, in contrast to the dynamic evaluation procedure in [15], and has a lower computational time (see section 6), compared with the de Casteljau-like B-algorithm [1,7,8,9] (analogue of the de Casteljau algorithm for classical polynomial Bézier curves), and with the algorithm introduced by Woźny and Chudy in [13].…”