A Subdivision Arrangement Algorithm for Semi-Algebraic Curves: An Overview

We overview a new method for computing the arrangement of semi-algebraic curves. A subdivision approach is used to compute the topology of the algebraic objects and to segment the boundary of regions defined by these objects. An efficient insertion technique is described, which detects regions in conflict and updates the underlying arrangement structure. We describe the general framework of this method, the main region insertion operation and the specializations of the key ingredients for the different types of objects: implicit, parametric or piecewise linear curves.

[1]  Nicola Wolpert,et al.  Jacobi Curves : Computing the Exact Topology of Arrangements of Non-Singular Algebraic Curves , 2000 .

[2]  Micha Sharir,et al.  Arrangements and Their Applications , 2000, Handbook of Computational Geometry.

[3]  Nira Dyn,et al.  A 4-point interpolatory subdivision scheme for curve design , 1987, Comput. Aided Geom. Des..

[4]  Tony DeRose,et al.  Efficient, fair interpolation using Catmull-Clark surfaces , 1993, SIGGRAPH.

[5]  Gilbert Strang,et al.  Wavelets and Dilation Equations: A Brief Introduction , 1989, SIAM Rev..

[6]  C. Micchelli,et al.  On vector subdivision , 1998 .

[7]  J. Boissonnat,et al.  Algorithmic Geometry: Frontmatter , 1998 .

[8]  Thomas Ottmann,et al.  Algorithms for Reporting and Counting Geometric Intersections , 1979, IEEE Transactions on Computers.

[9]  Mariette Yvinec,et al.  Algorithmic geometry , 1998 .

[10]  Bernard Mourrain,et al.  Subdivision methods for solving polynomial equations , 2009, J. Symb. Comput..

[11]  D. Levin,et al.  Analysis of quasi-uniform subdivision , 2003 .

[12]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[13]  Gershon Elber,et al.  Geometric constraint solver using multivariate rational spline functions , 2001, SMA '01.

[14]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[15]  Irina Voiculescu,et al.  Interval methods in geometric modeling , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[16]  Victor J. Milenkovic,et al.  An approximate arrangement algorithm for semi-algebraic curves , 2006, SCG '06.

[17]  Leif Kobbelt,et al.  Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology , 1996, Comput. Graph. Forum.

[18]  I. Emiris,et al.  Real Algebraic Numbers: Complexity Analysis and Experimentations , 2008 .

[19]  Gershon Elber,et al.  Contouring 1- and 2-manifolds in arbitrary dimensions , 2005, International Conference on Shape Modeling and Applications 2005 (SMI' 05).

[20]  Ulrich Reif,et al.  A unified approach to subdivision algorithms near extraordinary vertices , 1995, Comput. Aided Geom. Des..

[21]  Jürgen Garloff,et al.  Investigation of a subdivision based algorithm for solving systems of polynomial equations , 2001 .

[22]  Peter Schröder,et al.  Interpolating Subdivision for meshes with arbitrary topology , 1996, SIGGRAPH.

[23]  Nicholas M. Patrikalakis,et al.  Computation of the solutions of nonlinear polynomial systems , 1993, Comput. Aided Geom. Des..

[24]  Lyle Ramshaw,et al.  Blossoms are polar forms , 1989, Comput. Aided Geom. Des..

[25]  Jörg Peters,et al.  A realtime GPU subdivision kernel , 2005, SIGGRAPH 2005.

[26]  Bernard Mourrain,et al.  On the computation of an arrangement of quadrics in 3D , 2005, Comput. Geom..