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

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

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Citation

Wolpert, N. (2002). An Exact and Efficient Approach for Computing a Cell in an Arrangement of Quadrics. PhD Thesis, Universität des Saarlandes, Saarbrücken.


Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-2EFC-A
Abstract
In this thesis, 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 can be 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 {\em Jacobi curve}, we are able to find non-singular tangential intersections. By a generalization of the Jacobi curve we are able to determine non-singular tangential intersections in arbitrary planar arrangements. 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.