On the Different Shapes Arising in a Family of Rational Curves Depending on a Parameter

TL;DR

This paper presents an algorithm that analyzes how the shape of a family of rational curves varies with a parameter, enabling the determination of topology types without computing implicit equations.

Contribution

The paper introduces a parametric algorithm for classifying the shape of rational curves depending on a parameter, improving efficiency over implicit methods.

Findings

The parametric algorithm outperforms implicit methods in timing.

The algorithm accurately partitions the parameter space by shape.

Topology types of the family can be determined from the partition.

Abstract

Given a family of rational curves depending on a real parameter, defined by its parametric equations, we provide an algorithm to compute a finite partition of the parameter space (${\Bbb R}$, in general) so that the shape of the family stays invariant along each element of the partition. So, from this partition the topology types in the family can be determined. The algorithm is based on a geometric interpretation of previous work (\cite{JGRS}) for the implicit case. However, in our case the algorithm works directly with the parametrization of the family, and the implicit equation does not need to be computed. Timings comparing the algorithm in the implicit and the parametric cases are given; these timings show that the parametric algorithm developed here provides in general better results than the known algorithm for the implicit case.