A geometric characterization of parametric cubic curves
ACM Transactions on Graphics (TOG)
Detecting cusps and inflection points in curves
Computer Aided Geometric Design
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
A polynomial approach to linear algebra
A polynomial approach to linear algebra
Identification of inflection points and cusps on rational curves
Computer Aided Geometric Design
The moving line ideal basis of planar rational curves
Computer Aided Geometric Design
Inflection points and singularities on planar rational cubic curve segments
Computer Aided Geometric Design
On the minors of the implicitization Bézout matrix for a rational plane curve
Computer Aided Geometric Design
A new implicit representation of a planar rational curve with high order singularity
Computer Aided Geometric Design
The µ-basis of a planar rational curve: properties and computation
Graphical Models
Elimination Practice: Software Tools and Applications
Elimination Practice: Software Tools and Applications
Computation of the singularities of parametric plane curves
Journal of Symbolic Computation
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
μ-Bases and singularities of rational planar curves
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
Matrix-based implicit representations of rational algebraic curves and applications
Computer Aided Geometric Design
Journal of Symbolic Computation
Using Smith normal forms and µ-bases to compute all the singularities of rational planar curves
Computer Aided Geometric Design
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We compute the singular points of a plane rational curve, parametrically given, using the implicitization matrix derived from the @m-basis of the curve. It is shown that singularity factors, which are defined and uniquely determined by the elementary divisors of the implicitization matrix, contain all the information about the singular points, such as the parameter values of the singular points and their multiplicities. Based on this observation, an efficient and numerically stable algorithm for computing the singular points is devised, and inversion formulae for the singular points are derived. In particular, high order singular points can be detected and computed effectively. This approach based on singularity factors can also determine whether a rational curve has any non-ordinary singular points that contain singular points in its infinitely near neighborhood. Furthermore, a method is proposed to determine whether a singular point is ordinary or not. Finally, a conjecture in [Chionh, E.-W., Sederberg, T.W., 2001. On the minors of the implicitization bezout matrix for a rational plane curve. Computer Aided Geometric Design 18, 21-36] regarding the multiplicity of the singular points of a plane rational curve is proved.