Isotopic Arrangement of Simple Curves: An Exact Numerical Approach Based on Subdivision

We present a purely numerical (i.e., non-algebraic) subdivision algorithm for computing an isotopic approximation of a simple arrangement of curves. The arrangement is “simple” in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function on the plane, along with effective interval forms of the function and its partial derivatives. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A preliminary implementation is available in Core Library.

[1]  Chee-Keng Yap.  Complete subdivision algorithms, I: intersection of Bezier curves , 2006, SCG '06.

[2]  Jean-Daniel Boissonnat,et al.  Effective computational geometry for curves and surfaces , 2006 .

[3]  Bernard Mourrain,et al.  Continued fraction expansion of real roots of polynomial systems , 2009, SNC '09.

[4]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[5]  Sylvain Pion,et al.  Interval arithmetic yields efficient dynamic filters for computational geometry , 1998, SCG '98.

[6]  Jihun Yu,et al.  The Design of Core 2: A Library for Exact Numeric Computation in Geometry and Algebra , 2010, ICMS.

[7]  Bernard Mourrain,et al.  On continued fraction expansion of real roots of polynomial systems, complexity and condition numbers , 2011, Theor. Comput. Sci..

[8]  Suhas N. Diggavi,et al.  Efficient Max-Norm Distance Computation for Reliable Voxelization , 2003, Symposium on Geometry Processing.

[9]  V. Stahl.  Interval Methods for Bounding the Range of Polynomials and Solving Systems of Nonlinear Equations , 2007 .

[10]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[11]  Ramon E. Moore,et al.  A Simple Test for Accuracy of Approximate Solutions to Nonlinear (or Linear) Systems , 1980 .

[12]  Simon Plantinga,et al.  Certified algorithms for implicit surfaces , 2007 .

[13]  Gert Vegter,et al.  Isotopic approximation of implicit curves and surfaces , 2004, SGP '04.

[14]  Wladyslaw Kulpa,et al.  THE POINCARE-MIRANDA THEOREM , 1997 .

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

[16]  Narayan Kamath,et al.  Subdivision Algorithms for Complex Root Isolation: Empirical Comparisons , 2010 .

[17]  Chee-Keng Yap,et al.  A simple but exact and efficient algorithm for complex root isolation , 2011, ISSAC '11.

[18]  Chee-Keng Yap,et al.  Adaptive isotopic approximation of nonsingular curves: the parametrizability and nonlocal isotopy approach , 2009, SCG '09.

[19]  Bruno Lang,et al.  Existence Tests for Solutions of Nonlinear Equations Using Borsuk's Theorem , 2005, SIAM J. Numer. Anal..

[20]  Chee K. Yap,et al.  Adaptive Isotopic Approximation of Nonsingular Curves and Surfaces , 2011 .

[21]  Chee-Keng Yap,et al.  Complete subdivision algorithms, II: isotopic meshing of singular algebraic curves , 2008, ISSAC '08.

[22]  Hong Yi,et al.  A survey of the marching cubes algorithm , 2006, Comput. Graph..

[23]  Chee-Keng Yap,et al.  Adaptive Isotopic Approximation of Nonsingular Curves: the Parameterizability and Nonlocal Isotopy Approach , 2011, Discret. Comput. Geom..

[24]  Gunter Rote.  Extension of geometric filtering techniques to higher-degree parametric curves , 2006 .

[25]  Stefan Schirra,et al.  In Praise of Numerical Computation , 2010 .

[26]  Dinesh Manocha,et al.  Topology preserving surface extraction using adaptive subdivision , 2004, SGP '04.