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Homotopy Conditions for Tolerant Geometric Queries

  • Conference paper
Reliable Implementation of Real Number Algorithms: Theory and Practice

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5045))

Abstract

Algorithms for many geometric queries rely on representations that are comprised of combinatorial (logical, incidence) information, usually in a form of a graph or a cell complex, and geometric data that represents embeddings of the cells in the Euclidean space E d. Whenever geometric embeddings are imprecise, their incidence relationships may become inconsistent with the associated combinatorial model. Tolerant algorithms strive to compute on such representations despite the inconsistencies, but the meaning and correctness of such computations have been a subject of some controversy.

This paper argues that a tolerant algorithm usually assumes that the approximate geometric representation corresponds to a subset of E d that is homotopy equivalent to the intended exact set. We show that the Nerve Theorem provides systematic means for identifying sufficient conditions for the required homotopy equivalence, and explain how these conditions are used in the context of geometric and solid modeling.

Based on the talk at the Dagstuhl Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice, January 8-13, 2006.

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Authors and Affiliations

  1. Mechanical Engineering and Computer Sciences, University of Wisconsin, Madison, Wisconsin, 53706, USA

    Vadim Shapiro

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  1. Vadim Shapiro
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Peter Hertling Christoph M. Hoffmann Wolfram Luther Nathalie Revol

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Shapiro, V. (2008). Homotopy Conditions for Tolerant Geometric Queries. In: Hertling, P., Hoffmann, C.M., Luther, W., Revol, N. (eds) Reliable Implementation of Real Number Algorithms: Theory and Practice. Lecture Notes in Computer Science, vol 5045. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85521-7_10

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  • DOI: https://doi.org/10.1007/978-3-540-85521-7_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85520-0

  • Online ISBN: 978-3-540-85521-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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