Adaptive Approximate Implicitization of Planar Parametric Curves via Asymmetric Gradient Constraints

Abstract

Converting a parametric curve into the implicit form, which is called implicitization, has always been a popular but challenging problem in geometric modeling and related applications. However, existing methods mostly suffer from the problems of maintaining geometric features and choosing a reasonable implicit degree. The present paper has two contributions. We first introduce a new regularization constraint (called the asymmetric gradient constraint) for both polynomial and non-polynomial curves, which efficiently possesses shape-preserving. We then propose two adaptive algorithms of approximate implicitization for polynomial and non-polynomial curves respectively, which find the “optimal” implicit degree based on the behavior of the asymmetric gradient constraint. More precisely, the idea is gradually increasing the implicit degree, until there is no obvious improvement in the asymmetric gradient loss of the outputs. Experimental results have shown the effectiveness and high quality of our proposed methods.