Topology of 2D and 3D Rational Curves

TL;DR

This paper introduces algorithms for determining the topology of 2D and 3D rational curves directly from their parametrizations, avoiding implicit equations, with demonstrated good performance in Maple implementations.

Contribution

It presents novel algorithms that compute the topology of rational curves directly from parametrizations without relying on implicit equations.

Findings

Algorithms work directly with parametrizations

Implemented in Maple with good performance

Applicable to both planar and space rational curves

Abstract

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or use the implicit equation of the curve (in the case of planar curves) or of any projection (in the case of space curves). Moreover, these algorithms have been implemented in Maple; the examples considered and the timings obtained show good performance skills.