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Computing a 3-dimensional Cell in an Arrangement of Quadrics: Exactly and Actually!

MPS-Authors

Geismann,  Nicola
Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons44609

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

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

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

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Citation

Geismann, N., Hemmer, M., & Schömer, E. (2001). Computing a 3-dimensional Cell in an Arrangement of Quadrics: Exactly and Actually! In Proceedings of the 17th Annual Symposium on Computational Geometry (SCG-01) (pp. 264-273). New York: ACM.


Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-317E-3
Abstract
We present two approaches to the problem of calculating a cell in a 3-dimensional arrangement of quadrics. The first approach solves the problem using rational arithmetic. It works with reductions to planar arrangements of algebraic curves. Degenerate situations such as tangential intersections and self-intersections of curves are intrinsic to the planar arrangements we obtain. The coordinates of the intersection points are given by the roots of univariate polynomials. We succeed in locating all intersection points either by extended local box hit counting arguments or by globally characterizing them with simple square root expressions. The latter is realized by a clever factorization of the univariate polynomials. Only the combination of these two results facilitates a practical and implementable algorithm. The second approach operates directly in 3-space by applying classical solid modeling techniques. Whereas the first approach guarantees a correct solution in every case the second one may fail in some degenerate situations. But with the help of verified floating point arithmetic it can detect these critical cases and is faster if the quadrics are in general position.