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Implicitization via the Gröbner Walk

Daniel Lichtblau
Organization: Wolfram Research, Inc.

ACA 2007
Conference location

Oakland University, Rochester, MI

Using the Gröbner Walk to compute implicit equations defining parametrized curves and surfaces.

Talk given at the Conference on Applications of Computer Algebra (ACA) 2007, Approximate Algebraic Computation Session (July, 2007, Oakland University, Rochester, MI).


The Gröbner walk is a useful method for conversion from a "simple" Gröbner basis to a different one in a desired term order.Various issues along the way include coefficient swell (similar to that seen in the classical Buchberger algorithm), polynomials with many initial elements in the "end-game" phase, and the like.We take as benchmark the implicitization of the 32 bicubic parametric patches in the Newell (Utah) teapot.We will see how the Gröbner walk can be used to implicitize all of them, using in some cases approximate arithmetic and an early abort strategy, with a result that can be certified a posteriori.To the best of my knowledge the four spout patches have never before been amenable to a Gröbner basis approach. We show some other nontrivial examples from the implicitization literature.

*Mathematics > Algebra > Field and Ring Theory
*Mathematics > Algebra > Polynomials
*Mathematics > Geometry > Computational Geometry
*Mathematics > Geometry > Surfaces

implicitization, parametric equations, polynomial algebra, commutative algebra, computational algebraic geometry, nonlinear computational geometry, computer-aided geometric design, Gröbner bases, Gröbner walk

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ACA2007_GroebnerWalk.pdf (2.5 MB) - PDF Document
ACA2007_GroebnerWalk.nb (1.9 MB) - Mathematica Notebook