An Evolution-Based Approach for Approximate Parameterization of Implicitly Defined Curves by Polynomial Parametric Spline Curves

Abstract

Abstract. We propose a novel approach for the approximate parameterization of an implicitly defined curve in the plane by polynomial parametric spline curves. The method generates the parameterization of the curve (which may consist of several open and closed branches) without using any a priori information about its topology. If needed the topology of the approximate parameterization can be certified against the initial curve in a simple way.

Efficient and Validated Numerical Evaluation of Abelian Integrals

“…However, existence theorems for exact (polynomial, rational or trigonometric) parameterizations are mostly restricted to curves of genus 0 [Abhyankar and Bajaj 1988;Hong and Schicho 1998], while approximate methods (piecewise rational, splines, etc.) developed for more general curves most of the time come without guaranteed error bounds (see e.g., [Bajaj and Xu 1997;Gao and Li 2004;Yang et al 2010]). These methods usually target the standard binary64 precision (i.e.…”

A Hamiltonian approach to implicit systems, generalized solutions and applications in optimization

“…We also recall that, in algebraic geometry, implicitization and parametrization (via rational functions) are important subjects, Gao [13], Wang [37], Schicho [30]. General parametrization methods are not known, Gao [13] and recent papers study approximate parametrization approaches, Dobiasova [11], Yang, Jüttler and Gonzales-Vega [38].…”