Approximate algebraic methods for curves and surfaces and their applications

Jüttler, Bert; Chalmovianský, Pavel; Shalaby, Mohamed; Wurm, Elmar

doi:10.1145/1090122.1090124

Cited by 6 publications

(2 citation statements)

References 12 publications

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“…Ref. [21] addresses this problem by combining two or more eigenvectors (associated with small eigenvalues), which leads to a gradual decline in accuracy. With the AGM developed in this paper, additional branches can be avoided as much as possible in the implicitization procedure.…”

Section: Methodsmentioning

confidence: 99%

Adaptive Approximate Implicitization of Planar Parametric Curves via Asymmetric Gradient Constraints

Guo,

Gao

2023

Symmetry

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Converting a parametric curve into the implicit form, which is called implicitization, has always been a popular but challenging problem in geometric modeling and related applications. However, existing methods mostly suffer from the problems of maintaining geometric features and choosing a reasonable implicit degree. The present paper has two contributions. We first introduce a new regularization constraint (called the asymmetric gradient constraint) for both polynomial and non-polynomial curves, which efficiently possesses shape-preserving. We then propose two adaptive algorithms of approximate implicitization for polynomial and non-polynomial curves respectively, which find the “optimal” implicit degree based on the behavior of the asymmetric gradient constraint. More precisely, the idea is gradually increasing the implicit degree, until there is no obvious improvement in the asymmetric gradient loss of the outputs. Experimental results have shown the effectiveness and high quality of our proposed methods.

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Section: Methodsmentioning

confidence: 99%

Adaptive Approximate Implicitization of Planar Parametric Curves via Asymmetric Gradient Constraints

Guo,

Gao

2023

Symmetry

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“…Of these, the parametric form has established itself as the representation of choice in most CAGD systems due to its intuitive geometric nature [9]. However, the implicit form has distinct advantages over the parametric form in solving certain geometrical problems and thus the possibility to have a dual representation is, in some circumstances, appealing [10]. For example, the implicit representation allows us to immediately determine whether a given point lies on the curve or surface.…”

Section: Introductionmentioning

confidence: 99%

Approximate implicitization of triangular Bézier surfaces

Barrowclough

Dokken

2010

Proceedings of the 26th Spring Conference on Computer Graphics

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Abstract. We discuss how Dokken's methods of approximate implicitization can be applied to triangular Bézier surfaces in both the original and weak forms. The matrices D and M that are fundamental to the respective forms of approximate implicitization are shown to be constructed essentially by repeated multiplication of polynomials and by matrix multiplication. A numerical approach to weak approximate implicitization is also considered and we show that symmetries within this algorithm can be exploited to reduce the computation time of M. Explicit examples are presented to compare the methods and to demonstrate properties of the approximations.

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Geometric Modeling and Algebraic Geometry

Jttler¹,

Piene²

2008

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Approximate algebraic methods for curves and surfaces and their applications

Cited by 6 publications

References 12 publications

Adaptive Approximate Implicitization of Planar Parametric Curves via Asymmetric Gradient Constraints

Adaptive Approximate Implicitization of Planar Parametric Curves via Asymmetric Gradient Constraints

Approximate implicitization of triangular Bézier surfaces

Geometric Modeling and Algebraic Geometry

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