Curve implicitization using moving lines
Computer Aided Geometric Design
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
A direct approach to computing the &mgr;-basis of planar rational curves
Journal of Symbolic Computation
Reparametrization of a rational ruled surface using the μ-basis
Computer Aided Geometric Design
Reparametrization of a rational ruled surface using the μ-basis
Computer Aided Geometric Design
Computing μ-bases of rational curves and surfaces using polynomial matrix factorization
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Axial moving lines and singularities of rational planar curves
Computer Aided Geometric Design
Real implicitization of curves and geometric extraneous components
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Computing singular points of plane rational curves
Journal of Symbolic Computation
Division algorithms for Bernstein polynomials
Computer Aided Geometric Design
The implicit equation of a canal surface
Journal of Symbolic Computation
μ-bases for polynomial systems in one variable
Computer Aided Geometric Design
Computing self-intersection curves of rational ruled surfaces
Computer Aided Geometric Design
Axial moving planes and singularities of rational space curves
Computer Aided Geometric Design
A computational study of ruled surfaces
Journal of Symbolic Computation
Implicitization of rational ruled surfaces with µ-bases
Journal of Symbolic Computation
μ-Bases and singularities of rational planar curves
Computer Aided Geometric Design
The μ-basis and implicitization of a rational parametric surface
Journal of Symbolic Computation
Computing the singularities of rational space curves
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Approximate µ-bases of rational curves and surfaces
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Using Smith normal forms and µ-bases to compute all the singularities of rational planar curves
Computer Aided Geometric Design
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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A moving line L(x,y;t) = 0 is a family of lines with one parameter t in a plane. A moving line L(x,y;t) = 0 is said to follow a rational curve P ( t ) if the point P ( t 0 ) is on the line L (x, y; t 0 ) = 0 for any parameter value t 0 . A µ-basis of a rational curve P ( t ) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P ( t ), which is the Syzygy module of P ( t ). The study of moving lines, especially the µ-basis, has recently led to an efficient method, called the moving line method , for computing the implicit equation of a rational curve [3,6]. In this paper, we present properties and equivalent definitions of a µ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the µ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a µ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P ( t ), and has O( n 2 ) time complexity, where n is the degree of P ( t ). We show that the new algorithm is more efficient than the fastest previous algorithm [7].