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Journal for Geometry and Graphics 09 (2005), No. 1, 011--029
Copyright Heldermann Verlag 2005



Comparison of Implicitization Methods

Bohumir Bastl
Dept. of Mathematics, University of West Bohemia, UniversitnÝ 22, 306 14 Plzen, Czech Republic
bastl@kma.zcu.cz

Frantisek Jezek
Dept. of Mathematics, University of West Bohemia, UniversitnÝ 22, 306 14 Plzen, Czech Republic



This paper overviews several methods for implicitization of an algebraic variety (finding an implicit representation of a rational algebraic variety given by its parametric equations): the classical implicitization using Gr÷bner bases, the implicitization using resultants, polynomial interpolation, or moving curves and surfaces, and the direct implicitization method. All these methods are used for finding the implicit equation of NURBS curves and surfaces, which are typical objects in geometric modeling. At the end, the computational costs for finding the implicit equation needed by implementations of these different methods are compared.

Keywords: Implicitization methods, NURBS curves and surfaces.

MSC: 68W30; 13P10, 14Q05, 14Q10

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