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

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.

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 Creators:
Geismann, Nicola1, Author
Hemmer, Michael2, Author           
Schömer, Elmar2, Author           
Affiliations:
1Max Planck Society, ou_persistent13              
2Algorithms and Complexity, MPI for Informatics, Max Planck Society, ou_24019              

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 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.

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Language(s): eng - English
 Dates: 2010-03-022001
 Publication Status: Issued
 Pages: -
 Publishing info: New York : ACM
 Table of Contents: -
 Rev. Type: -
 Identifiers: eDoc: 518207
Other: Local-ID: C1256428004B93B8-DAEA50F63741FF02C1256B7B004AC4FE-ghs-qsi-01
 Degree: -

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Title: Untitled Event
Place of Event: Boston, Massachusetts
Start-/End Date: 2001

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Title: Proceedings of the 17th Annual Symposium on Computational Geometry (SCG-01)
Source Genre: Proceedings
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Publ. Info: New York : ACM
Pages: - Volume / Issue: - Sequence Number: - Start / End Page: 264 - 273 Identifier: -