PTOPO: Computing the Geometry and the Topology of Parametric Curves
Christina Katsamaki, Fabrice Rouillier, Elias Tsigaridas

TL;DR
The paper introduces PTOPO, an efficient algorithm for computing the topology and geometry of parametric curves directly in parameter space, improving complexity bounds and avoiding implicitization.
Contribution
PTOPO is a novel algorithm that computes the topology of parametric curves with improved complexity, operating in parameter space rather than implicit form.
Findings
Matches the best known complexity bounds for plane algebraic curves.
Improves the complexity for space curves by a factor of N^{10}.
Successfully implemented and tested in Maple for practical use.
Abstract
We consider the problem of computing the topology and describing the geometry of a parametric curve in . We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most and maximum bitsize of coefficients , then the worst case bit complexity of PTOPO is . This bound matches the current record bound for the problem of computing the topology of a plane algebraic curve given in implicit form. For plane and space curves, if $N…
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Taxonomy
TopicsPolynomial and algebraic computation · Computational Geometry and Mesh Generation · Advanced Combinatorial Mathematics
