Curve implicitization using moving lines
Computer Aided Geometric Design
Rational-ruled surfaces: implicitization and section curves
Graphical Models and Image Processing
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Implicitizing rational curves by the method of moving algebraic curves
Journal of Symbolic Computation - Special issue: parametric algebraic curves and applications
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
On the validity of implicitization by moving quadrics for rational surfaces with no base points
Journal of Symbolic Computation
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
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
Implicitization and parametrization of quadratic surfaces with one simple base point
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Computing self-intersection curves of rational ruled surfaces
Computer Aided Geometric Design
Minimal generators of the defining ideal of the Rees Algebra associated to monoid parameterizations
Computer Aided Geometric Design
Computing the singularities of rational space curves
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Computer Aided Geometric Design
Approximate µ-bases of rational curves and surfaces
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Journal of Symbolic Computation
Implicitizing rational surfaces of revolution using µ-bases
Computer Aided Geometric Design
Using µ-bases to implicitize rational surfaces with a pair of orthogonal directrices
Computer Aided Geometric Design
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The concept of a @m-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The @m-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. Furthermore, it can be used for surface reparametrization and computation of singular points. In this paper, we generalize the notion of a @m-basis to an arbitrary rational parametric surface. We show that: (1) the @m-basis of a rational surface always exists, the geometric significance of which is that any rational surface can be expressed as the intersection of three moving planes without extraneous factors; (2) the @m-basis is in fact a basis of the moving plane module of the rational surface; and (3) the @m-basis is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the @m-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Grobner basis. Problems for further research are also discussed. is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the @m-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Grobner basis. Problems for further research are also discussed.