Computing the topology of three-dimensional algebraic curves

Grégory Gatelier 1 Abder Labrouzy 1 Bernard Mourrain 1 Jean-Pierre Técourt 1
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this report, we present a new method for computing the topology of curves defined as the intersection of two implicit surfaces. The main ingredients are projection tools, based on resultant constructions and 0-dimensional polynomial system solvers. We describe a lifting method for points on the projection of the curve on a plane, even in the case of multiple preimages on the 3D curve. Reducing the problem to the comparison of coordinates of so-called critical points, we propose an approach which combines control and efficiency. An emphasis in this work is put on the experimental validation on this new method. Examples treated with the tools of the library (Algebraic Software-Components for gEometric modeLing) are showing the potential of such techniques.
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Grégory Gatelier, Abder Labrouzy, Bernard Mourrain, Jean-Pierre Técourt. Computing the topology of three-dimensional algebraic curves. [Research Report] RR-5194, INRIA. 2004, pp.21. ⟨inria-00070798⟩

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