On the Geometry and the Topology of Parametric Curves

Christina Katsamaki (1) , Fabrice Rouillier (1) , Elias Tsigaridas (1) , Zafeirakis Zafeirakopoulos (2)

1 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs
2 Gebze Technical University

We consider the problem of computing the topology and describing the geometry of a parametric curve in R. 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 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 O (6 + 5 + 4 (2 +) + 3 (2 + 3) + 3 2). This bound matches the current record bound O (6 + 5) for the problem of computing the topology of a planar algebraic curve given in implicit form. For planar and space curves, if = max{ , }, the complexity of PTOPO becomes O (6), which improves the state-of-the-art result, due to Alcázar and Díaz-Toca [CAGD'10], by a factor of 10. However, visualizing the curve on top of the abstract graph construction, increases the bound to O (7). We have implemented PTOPO in maple for the case of planar curves. Our experiments illustrate its practical nature.

Mots clés: Parametric curve; Topology; Bit complexity; Polynomial systems

Domaines: Calcul formel [cs.SC]; Géométrie algorithmique [cs.CG]

Christina Katsamaki, Fabrice Rouillier, Elias Tsigaridas, Zafeirakis Zafeirakopoulos. On the Geometry and the Topology of Parametric Curves. ISSAC 2020 - International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata / Virtual, Greece. ⟨hal-02573423v2⟩