Fast and exact geometric analysis of real algebraic plane curves

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Fast and exact geometric analysis of real algebraic plane curves

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International Conference on Symbolic and Algebraic Computation archive
Proceedings of the 2007 international symposium on Symbolic and algebraic computation table of contents

Waterloo, Ontario, Canada

SESSION: Contributed papers table of contents

Pages: 151 - 158

Year of Publication: 2007

ISBN:978-1-59593-743-8

Authors

Arno Eigenwillig	Max-Planck-Institut für Informatik
Michael Kerber	Max-Planck-Institut für Informatik
Nicola Wolpert	Hochschule für Technik

Sponsors

ACM: Association for Computing Machinery
SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 6, Downloads (12 Months): 31, Citation Count: 15

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ABSTRACT

An algorithm is presented for the geometric analysis of an algebraic curve f(x, y) = 0 in the real affine plane. It computes a cylindrical algebraic decomposition (CAD) of the plane, augmented with adjacency information. The adjacency information describes the curve's topology by a topologically equivalent planar graph. The numerical data in the CAD gives an embedding of the graph.

The algorithm is designed to provide the exact result for all inputs but to perform only few symbolic operations for the sake of efficiency. In particular, the roots of f(∝, y) at a critical x-coordinate .

The algorithm is implemented as C++ library AlciX in the EXACUS project. Running time comparisons with top by Gonzalez-Vega and Necula (2002), and with cad2d by Brown demonstrate its efficiency.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

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	Mhammed El Kahoui, Topology of real algebraic space curves, Journal of Symbolic Computation, v.43 n.4, p.235-258, April, 2008
	D. A. Aruliah , Robert M. Corless , Azar Shakoori , Laureano Gonzalez-Vega , Ignacio F. Rua, Computing the topology of a real algebraic plane curve whose equation is not directly available, Proceedings of the 2007 international workshop on Symbolic-numeric computation, July 25-27, 2007, London, Ontario, Canada
	Dimitrios I. Diochnos , Ioannis Z. Emiris , Elias P. Tsigaridas, On the complexity of real solving bivariate systems, Proceedings of the 2007 international symposium on Symbolic and algebraic computation, July 29-August 01, 2007, Waterloo, Ontario, Canada
	Jorge Caravantes , Laureano Gonzalez-Vega, Improving the topology computation of an arrangement of cubics, Computational Geometry: Theory and Applications, v.41 n.3, p.206-218, November, 2008
	Eric Berberich , Michael Sagraloff, A generic and flexible framework for the geometrical and topological analysis of (algebraic) surfaces, Computer Aided Geometric Design, v.26 n.6, p.627-647, August, 2009
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	Jinsan Cheng , Sylvain Lazard , Luis Peñaranda , Marc Pouget , Fabrice Rouillier , Elias Tsigaridas, On the topology of planar algebraic curves, Proceedings of the 25th annual symposium on Computational geometry, June 08-10, 2009, Aarhus, Denmark