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An Exact and Efficient Approach for Computing a Cell in an Arrangement of Quadrics

MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons45414

Schömer,  Elmar
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45758

Wolpert,  Nicola
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Zitation

Schömer, E., & Wolpert, N. (2006). An Exact and Efficient Approach for Computing a Cell in an Arrangement of Quadrics. Computational Geometry, 33, 65-97.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-000F-2208-A
Zusammenfassung
We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to the one of computing planar arrangements of algebraic curves. We succeed in locating all event points in these arrangements, including tangential intersections and singular points. By introducing an additional curve, which we call the Jacobi curve, we are able to find non-singular tangential intersections. We show that the coordinates of the singular points in our special projected planar arrangements are roots of quadratic polynomials. The coefficients of these polynomials are usually rational and contain at most a single square root. A prototypical implementation indicates that our approach leads to good performance in practice.