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Conference Paper

Geometric Analysis of Algebraic Surfaces Based on Planar Arrangements

MPS-Authors
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Berberich,  Eric
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Kerber,  Michael
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

/persons/resource/persons45332

Sagraloff,  Michael
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Citation

Berberich, E., Kerber, M., & Sagraloff, M. (2008). Geometric Analysis of Algebraic Surfaces Based on Planar Arrangements. In S. Petitjean (Ed.), 24th European Workshop on Computational Geometry: collection of abstracts/EuroCG'08 (pp. 29-32). Vandoeuvre-lès-Nancy: INPL.


Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-1BCD-9
Abstract
We present a method to compute the exact topology of a real algebraic surface $S$, implicitly given by a polynomial $f \in \mathbb{Q}[x,y,z]$ of arbitrary degree $N$. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of $S$ including critical points. We use a projection approach, similar to Collins' cylindrical algebraic decomposition (cad). In comparison we reduce the number of output cells to $O(N^5)$ by constructing a special planar arrangement instead of a full cad in the projection plane. Furthermore, our approach applies numerical and combinatorial methods to minimize costly symbolic computations. The algorithm handles all sorts of degeneracies without transforming the surface into a generic position. We provide a complete \Cpp-implementation of the algorithm that shows good performance for many well-known examples from algebraic geometry.