The Bernstein Basis and its applications in solving geometric constraint systems

Sebti Foufou, Dominique Michelucci

    Research output: Contribution to journalArticle

    Abstract

    This article reviews the properties of Tensorial Bernstein Basis (TBB) and its usage, with interval analysis, for solving systems of nonlinear, univariate or multivariate equations resulting from geometric constraints. TBB are routinely used in computerized geometry for geometric modelling in CAD-CAM, or in computer graphics. They provide sharp enclosures of polynomials and their derivatives. They are used to reduce domains while preserving roots of polynomial systems, to prove that domains do not contain roots, and to make existence and uniqueness tests. They are compatible with standard preconditioning methods and fit linear programming techniques. However, current Bernstein-based solvers are limited to small algebraic systems. We present Bernstein polytopes and show how combining them with linear programming allows us to solve larger systems as well. The article also gives a generalization of Bernstein polytopes to higher degrees and a comparison of polytopes-based versus TBB-based polynomial bounds.

    Original languageEnglish (US)
    Pages (from-to)192-208
    Number of pages17
    JournalReliable Computing
    Volume17
    StatePublished - Dec 1 2012

    Fingerprint

    Geometric Constraint Solving
    Bernstein Basis
    Polytopes
    Polynomials
    Linear programming
    Roots
    CAD/CAM
    Interval Analysis
    Geometric Modeling
    Geometric Constraints
    Polynomial
    Polynomial Systems
    Enclosure
    Computer aided manufacturing
    Computer graphics
    Preconditioning
    Enclosures
    Univariate
    Computer aided design
    Existence and Uniqueness

    Keywords

    • Algebraic systems
    • Geometric constraint solving. bernstein polytope
    • Tensorial Bernstein Basis
    • Univariate and multivariate polynomials

    ASJC Scopus subject areas

    • Software
    • Computational Mathematics
    • Applied Mathematics

    Cite this

    The Bernstein Basis and its applications in solving geometric constraint systems. / Foufou, Sebti; Michelucci, Dominique.

    In: Reliable Computing, Vol. 17, 01.12.2012, p. 192-208.

    Research output: Contribution to journalArticle

    Foufou, Sebti ; Michelucci, Dominique. / The Bernstein Basis and its applications in solving geometric constraint systems. In: Reliable Computing. 2012 ; Vol. 17. pp. 192-208.
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