Moving curve ideals of rational plane parametrizations

TL;DR

This paper surveys methods for implicitizing rational plane parametrizations, discussing their mathematical foundations, current results, and open questions in the intersection of geometric design and algebraic geometry.

Contribution

It provides a comprehensive overview of existing techniques for implicitization, highlighting recent developments and open problems in the field.

Findings

Multiple methods analyzed for implicitization efficiency

Mathematical formulations of key techniques presented

Current results and open questions summarized

Abstract

In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the Computer Aided Geometric Design Community. The analysis of the validity of these techniques has been a fruitful ground for Commutative Algebraists and Algebraic Geometers, and several results have been obtained so far. Yet, a lot of research is still being done currently around this topic. In this note we present these methods, show their mathematical formulation, and survey current results and open questions.