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  Exact Arrangements on Tori and Dupin Cyclides

Berberich, E., & Kerber, M. (2008). Exact Arrangements on Tori and Dupin Cyclides. In E. Haines, & M. McGuire (Eds.), Proceedings SPM 2008: ACM Solid and Physical Modeling Symposium (pp. 59-66). New York, NY: ACM.

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Item Permalink: https://hdl.handle.net/11858/00-001M-0000-000F-1B91-D Version Permalink: https://hdl.handle.net/11858/00-001M-0000-0028-8A0D-4
Genre: Conference Paper

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 Creators:
Berberich, Eric1, Author           
Kerber, Michael1, Author           
Affiliations:
1Algorithms and Complexity, MPI for Informatics, Max Planck Society, ou_24019              

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 Abstract: An algorithm and implementation is presented to compute the exact arrangement induced by arbitrary algebraic surfaces on a parametrized ring Dupin cyclide. The family of Dupin cyclides contains as a special case the torus. The intersection of an algebraic surface of degree $n$ with a reference cyclide is represented as a real algebraic curve of bi-degree $(2n,2n)$ in the two-dimensional parameter space of the cyclide. We use Eigenwillig and Kerber: ``Exact and Efficient 2D-Arrangements of Arbitrary Algebraic Curves'', SODA~2008, to compute a planar arrangement of such curves and extend their approach to obtain more asymptotic information about curves approaching the boundary of the cyclide's parameter space. With that, we can base our implementation on the general software framework by Berberich~et.~al.: ``Sweeping and Maintaining Two-Dimensional Arrangements on Surfaces: A First Step'', ESA~2007. Our contribution provides the demanded techniques to model the special geometry of surfaces intersecting a cyclide and the special topology of the reference surface of genus one. The contained implementation is complete and does not assume generic position. Our experiments show that the combinatorial overhead of the framework does not harm the efficiency of the method. Our experiments show that the overall performance is strongly coupled to the efficiency of the implementation for arrangements of algebraic plane curves.

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Language(s): eng - English
 Dates: 2009-03-232008
 Publication Status: Issued
 Pages: -
 Publishing info: -
 Table of Contents: -
 Rev. Type: -
 Identifiers: eDoc: 427992
DOI: 10.1145/1364901.1364912
URI: http://doi.acm.org/10.1145/1364901.1364912
Other: Local-ID: C125756E0038A185-45FA1D784FCF4D89C12574720059FF46-bk-eatdc-08
 Degree: -

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Title: 2008 ACM Solid and Physical Modeling Symposium
Place of Event: Stony Brook, NY
Start-/End Date: 2008-06-02 - 2008-06-04

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Title: Proceedings SPM 2008 : ACM Solid and Physical Modeling Symposium
  Abbreviation : SPM 2008
Source Genre: Proceedings
 Creator(s):
Haines, Eric1, Editor
McGuire, Morgan1, Editor
Affiliations:
1 External Organizations, ou_persistent22            
Publ. Info: New York, NY : ACM
Pages: - Volume / Issue: - Sequence Number: - Start / End Page: 59 - 66 Identifier: ISBN: 978-1-60558-106-4