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Applications of Gröbner bases in non-linear computational geometry

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Trends in Computer Algebra

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 296))

Abstract

Gröbner bases are certain finite sets of multivariate polynomials. Many problems in polynomial ideal theory (algebraic geometry, non-linear computational geometry) can be solved by easy algorithms after transforming the polynomial sets involved in the specification of the problems into Gröbner basis form. In this paper we give some examples of applying the Gröbner bases method to problems in non-linear computational geometry (inverse kinematics in robot programming, collision detection for superellipsoids, implicitization of parametric representations of curves and surfaces, inversion problem for parametric representations, automated geometrical theorem proving, primary decomposition of implicitly defined geometrical objects). The paper starts with a brief summary of the Gröbner bases method.

RISC-LINZ (Research Institute for Symbolic Computation) Johannes Kepler University, A4040 Linz, Austria. This research is supported by a grant from VOEST-ALPINE, Linz, (Dipl. Ing. H. Exner), and a grant from SIEMENS, München, (Dr. H. Schwärtzel)

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Authors
  1. Bruno Buchberger
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Rainer Janßen

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© 1988 Springer-Verlag Berlin Heidelberg

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Buchberger, B. (1988). Applications of Gröbner bases in non-linear computational geometry. In: Janßen, R. (eds) Trends in Computer Algebra. Lecture Notes in Computer Science, vol 296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-18928-9_5

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  • DOI: https://doi.org/10.1007/3-540-18928-9_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18928-2

  • Online ISBN: 978-3-540-38850-0

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