Numerical proper reparametrization of parametric plane curves

We present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. More precisely, given a tolerance ϵ>0 and a rational parametrization P of a plane curve C with perturbed float coefficients, we present an algorithm that computes a parametrization Q of a new plane curve D such that Q is an ϵ –proper reparametrization of D. In addition, the error bound is carefully discussed and we present a formula that measures the “closeness” between the input curve C and the output curve D

Rational quadratic approximation to real algebraic curves

www.elsevier.com/locate/cagd An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic B-spline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The final approximation keeps many important geometric features of the original curve such as the topology, convexity and sharp points. Our method is implemented and experiments show that it may achieve better approximation bound with less segments than previously known methods. We also extend the method to approximate spatial algebraic curves

Abstract Rational Quadratic Approximation to Real Algebraic Curves ⋆

An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic B-spline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The final approximation keeps many important geometric features of the original curve such as the topology, convexity and sharp points. Our method is implemented and experiments show that it may achieve better approximation bound with less segments than previously known methods. We also extend the method to approximate spatial algebraic curves. Key words: real algebraic curve, parametrization, approximation, topology determination.