7001

Exact Symbolic-Numeric Computation of Planar Algebraic Curves

Eric Berberich, Pavel Emeliyanenko, Alexander Kobel, Michael Sagraloff
Max-Planck-Institut fur Informatik, Campus E1 4, D-66123 Saarbrucken, Germany
arXiv:1201.1548v1 [cs.CG] (7 Jan 2012)

@article{2012arXiv1201.1548B,

   author={Berberich}, E. and {Emeliyanenko}, P. and {Kobel}, A. and {Sagraloff}, M.},

   title={"{Exact Symbolic-Numeric Computation of Planar Algebraic Curves}"},

   journal={ArXiv e-prints},

   archivePrefix={"arXiv"},

   eprint={1201.1548},

   primaryClass={"cs.CG"},

   keywords={Computer Science – Computational Geometry, Computer Science – Mathematical Software, Computer Science – Symbolic Computation, Mathematics – Algebraic Geometry, Mathematics – Geometric Topology, 14Q05 (Primary), 14h50, 14P10 (Secondary), I.1.2, G.4, D.2.13},

   year={2012},

   month={jan},

   adsurl={http://adsabs.harvard.edu/abs/2012arXiv1201.1548B},

   adsnote={Provided by the SAO/NASA Astrophysics Data System}

}

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We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project CGAL. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, LGP and Maple’s Isolate for polynomial system solving, and CGAL’s bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.
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