Isotopic triangulation of a real algebraic surface

We present a new algorithm for computing the topology of a real algebraic surface S in a ball B, even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to S, and even a simplicial complex isotopic to S by exploiting properties of the contour curve of S. The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of S, by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of S from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper.

[1]  H. Hironaka.  Triangulations of algebraic sets , 1974 .

[2]  M. Goresky,et al.  Stratified Morse theory , 1988 .

[3]  Patrizia M. Gianni,et al.  Algorithms to compute the topology of orientable real algebraic surfaces , 2003, J. Symb. Comput..

[4]  V.Kharlamov,et al.  The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves , 2003, math/0301245.

[5]  M. Coste.  AN INTRODUCTION TO SEMIALGEBRAIC GEOMETRY , 2002 .

[6]  Alexandru Dimca,et al.  Singularities and Topology of Hypersurfaces , 1992 .

[7]  Michael Sagraloff,et al.  Exact geometric-topological analysis of algebraic surfaces , 2008, SCG '08.

[8]  Laureano González-Vega,et al.  Efficient topology determination of implicitly defined algebraic plane curves , 2002, Comput. Aided Geom. Des..

[9]  J. P. Speder,et al.  EQUISINGULARITE ET CONDITIONS DE WHITNEY. , 1975 .

[10]  Arjeh M. Cohen,et al.  Some tapas of computer algebra , 1999, Algorithms and computation in mathematics.

[11]  B. Mourrain,et al.  Meshing implicit algebraic surfaces : the smooth case , 2004 .

[12]  S. Yu. Orevkov,et al.  The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves , 2003, J. Comb. Theory, Ser. A.

[13]  George E. Collins,et al.  Local Box Adjacency Algorithms for Cylindrical Algebraic Decompositions , 2002, J. Symb. Comput..

[14]  M'hammed El Kahoui,et al.  Topology of real algebraic space curves , 2008, J. Symb. Comput..

[15]  T. Dokken,et al.  Computational Methods for Algebraic Spline Surfaces , 2008 .

[16]  Josef Schicho,et al.  A delineability-based method for computing critical sets of algebraic surfaces , 2007, J. Symb. Comput..

[17]  Robert Hardt,et al.  Triangulation of subanalytic sets and proper light subanalytic maps , 1976 .

[18]  Bernard Mourrain,et al.  Explicit factors of some iterated resultants and discriminants , 2006, Math. Comput..

[19]  Alexander Varchenko,et al.  The classification of critical points, caustics and wave fronts , 1985 .

[20]  J. Rafael Sendra,et al.  Computation of the topology of real algebraic space curves , 2005, J. Symb. Comput..

[21]  Bernard Mourrain,et al.  Visualisation of Implicit Algebraic Curves , 2007, 15th Pacific Conference on Computer Graphics and Applications (PG'07).

[22]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[23]  Daniel Lazard,et al.  Resolution des Systemes d'Equations Algebriques , 1981, Theor. Comput. Sci..

[24]  Oscar Zariski,et al.  Studies in Equisingularity II. Equisingularity in Codimension 1 (and Characteristic Zero) , 1965 .

[25]  V. Kharlamov,et al.  Asymptotic growth of the number of classes of real plane algebraic curves when the degree increases , 2000 .

[26]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[27]  B. Mourrain,et al.  Visualisation of Implicit Algebraic Curves , 2007 .

[28]  Fabrice Rouillier,et al.  Symbolic Recipes for Polynomial System Solving , 1999 .

[29]  Jean-Daniel Boissonnat,et al.  Isotopic Implicit Surface Meshing , 2004, STOC '04.

[30]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[31]  D. Trotman,et al.  On Canny's roadmap algorithm: orienteering in semialgebraic sets (an application of singularity theory to theoretical robotics) , 1991 .

[32]  Georges Comte,et al.  Tame Geometry with Application in Smooth Analysis , 2004, Lecture notes in mathematics.

[33]  Patrizia M. Gianni,et al.  Algorithmical determination of the topology of a real algebraic surface , 2004, J. Symb. Comput..

[34]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[35]  Bernard Mourrain,et al.  Computing the Topology of Three-Dimensional Algebraic Curves , 2005 .

[36]  Bernard Mourrain,et al.  On the computation of the topology of a non-reduced implicit space curve , 2008, ISSAC '08.

[37]  M-F Roy,et al.  Géométrie algébrique réelle , 1987 .