Computing singular points of plane rational curves

We compute the singular points of a plane rational curve, parametrically given, using the implicitization matrix derived from the μ-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 bézout 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.