On the isotopic meshing of an algebraic implicit surface

We present a new and complete algorithm for computing the topology of an algebraic surface S given by a square free polynomial in QX,Y,Z]. Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in Diatta et al. (2008), on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S. We exploit a simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in O?B(d21?) for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficient polynomial of degree d and coefficient size ?. Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces.

[1]  Michael Kerber,et al.  Fast and exact geometric analysis of real algebraic plane curves , 2007, ISSAC '07.

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

[3]  Scott McCallum,et al.  A polynomial-time algorithm for the topological type of a real algebraic curve , 1984 .

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

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

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

[7]  H. Hong.  An efficient method for analyzing the topology of plane real algebraic curves , 1996 .

[8]  T. Sakkalis.  The topological configuration of a real algebraic curve , 1991, Bulletin of the Australian Mathematical Society.

[9]  Bernard Mourrain,et al.  Topology and arrangement computation of semi-algebraic planar curves , 2008, Comput. Aided Geom. Des..

[10]  Chandrajit L. Bajaj,et al.  Tracing surface intersections , 1988, Comput. Aided Geom. Des..

[11]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[12]  Daniel Reischert.  Asymptotically fast computation of subresultants , 1997, ISSAC.

[13]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

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

[15]  J. Risler,et al.  Real algebraic and semi-algebraic sets , 1990 .

[16]  Dinesh Manocha,et al.  Efficient and accurate B-rep generation of low degree sculptured solids using exact arithmetic , 1997, SMA '97.

[17]  C. Gibson.  REAL ALGEBRAIC AND SEMI‐ALGEBRAIC SETS (Actualités Mathématiques 348) , 1991 .

[18]  Michael E. Mortenson,et al.  Geometric Modeling , 2008, Encyclopedia of GIS.

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

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

[21]  Franz Winkler,et al.  PLOTTING ALGEBRAIC SPACE CURVES BY CLUSTER COMPUTING , 2000 .

[22]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[23]  Dinesh Manocha,et al.  Eecient and Accurate B-rep Generation of Low Degree Sculptured Solids Using Exact Arithmetic:i - Representations , 1999 .

[24]  B. Mourrain,et al.  Meshing of Surfaces , 2006 .

[25]  Thomas A. Grandine,et al.  A new approach to the surface intersection problem , 1997, Comput. Aided Geom. Des..

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

[27]  Bernard Mourrain,et al.  Isotopic triangulation of a real algebraic surface , 2009, J. Symb. Comput..

[28]  Stefan Arnborg,et al.  Algebraic decomposition of regular curves , 1986, SYMSAC '86.

[29]  Michel Coste,et al.  Thom's Lemma, the Coding of Real Algebraic Numbers and the Computation of the Topology of Semi-Algebraic Sets , 1988, J. Symb. Comput..

[30]  V. Pan.  Optimal and nearly optimal algorithms for approximating polynomial zeros , 1996 .

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

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

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

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

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