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Regularity Criteria for the Topology of Algebraic Curves and Surfaces

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Mathematics of Surfaces XII (Mathematics of Surfaces 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4647))

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Abstract

In this paper, we consider the problem of analysing the shape of an object defined by polynomial equations in a domain. We describe regularity criteria which allow us to determine the topology of the implicit object in a box from information on the boundary of this box. Such criteria are given for planar and space algebraic curves and for algebraic surfaces. These tests are used in subdivision methods in order to produce a polygonal approximation of the algebraic curves or surfaces, even if it contains singular points. We exploit the representation of polynomials in Bernstein basis to check these criteria and to compute the intersection of edges or facets of the box with these curves or surfaces. Our treatment of singularities exploits results from singularity theory such as an explicit Whitney stratification or the local conic structure around singularities. A few examples illustrate the behavior of the algorithms.

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Authors and Affiliations

  1. GALAAD, INRIA, BP 93, 06902 Sophia Antipolis, France

    Lionel Alberti & Bernard Mourrain

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  1. Lionel Alberti
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  2. Bernard Mourrain
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Ralph Martin Malcolm Sabin Joab Winkler

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Alberti, L., Mourrain, B. (2007). Regularity Criteria for the Topology of Algebraic Curves and Surfaces. In: Martin, R., Sabin, M., Winkler, J. (eds) Mathematics of Surfaces XII. Mathematics of Surfaces 2007. Lecture Notes in Computer Science, vol 4647. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73843-5_1

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  • DOI: https://doi.org/10.1007/978-3-540-73843-5_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73842-8

  • Online ISBN: 978-3-540-73843-5

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