# Adaptive Approximate Implicitization of Planar Parametric Curves via Weak Gradient Constraints

## TL;DR

This paper introduces a novel regularization constraint called the weak gradient constraint for implicitizing planar parametric curves, enabling shape preservation and adaptive degree selection, improving the quality of implicitization.

## Contribution

It proposes a new weak gradient constraint and adaptive algorithms for approximate implicitization, addressing feature preservation and implicit degree selection.

## Key findings

- Effective shape preservation demonstrated
- Adaptive algorithms find optimal implicit degrees
- High-quality implicitization results achieved

## 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, the 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 weak 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 weak gradient constraint. More precisely, the idea is gradually increasing the implicit degree, until there is no obvious improvement in the weak gradient loss of the outputs. Experimental results have shown the effectiveness and high quality of our proposed methods.