Degree reduction of Be´zier curves
Computer-Aided Design
Degree reduction of Be´zier curves
Selected papers of the international symposium on Free-form curves and free-form surfaces
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
Approximate implicitization using monoid curves and surfaces
Graphical Models and Image Processing
The mu-basis of a rational ruled surface
Computer Aided Geometric Design
A direct approach to computing the &mgr;-basis of planar rational curves
Journal of Symbolic Computation
A new implicit representation of a planar rational curve with high order singularity
Computer Aided Geometric Design
Reparametrization of a rational ruled surface using the μ-basis
Computer Aided Geometric Design
The µ-basis of a planar rational curve: properties and computation
Graphical Models
Approximate implicitization via curve fitting
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Revisiting the µ-basis of a rational ruled surface
Journal of Symbolic Computation
Computing μ-bases of rational curves and surfaces using polynomial matrix factorization
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
The μ-basis and implicitization of a rational parametric surface
Journal of Symbolic Computation
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The μ-bases of rational curves and surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of curves and surfaces. However, exact μ-bases may have high degree with complicated rational coefficients and are often hard to compute (especially for surfaces), and sometimes they are not easy to use in geometric modeling and processing applications. In this paper, we introduce approximate μ-bases for rational curves and surfaces, and present an algorithm to compute approximate μ-bases. The algorithm amounts to solving a generalized eigenvalue problem and some quadratic programming problems with linear constraints. As applications, approximate implicitization and degree reduction of rational curves and surfaces with approximate μ-bases are discussed. Both the parametric equations and the implicit equations of the approximate curves/surfaces are easily obtained by using the approximate μ-bases. As indicated by the examples, the proposed algorithm may be a useful alternative to other methods for approximate implicitization.