Moving Curve Ideals of Rational Plane Parametrizations

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. 1. Rational Plane Curves Rational curves are fundamental tools in Computer Aided Geometric Design. They are used to trace the boundary of any kind of shape via transforming a parameter (a number) via some simple algebraic operations into a point of the cartesian plane or three-dimensional space. Precision and esthetics in Computer Graphics demands more and more sophisticated calculations, and hence any kind of simplification of the very large list of tasks that need to be performed between the input and the output is highly appreciated in this world. In this survey, we will focus on a simplification of a method for implicitization rational curves and surfaces defined parametrically. This method was developed in the 90’s by Thomas Sederberg and his collaborators(see [STD94, SC95, SGD97]), and turned out to become a very rich and fruitful area of interaction among mathematicians, engineers and computer scientist. As we will see at the end of the survey, it is still a very active of research these days. Figure 1. The shape of an “orange” plotted with Mathematica 8.0 ([Wol10]). 2010 Mathematics Subject Classification. Primary 14H50; Secondary 13A30, 68W30. Partially supported by the Research Project MTM2010–20279 from the Ministerio de Ciencia e Innovación, Spain.