Using a bihomogeneous resultant to find the singularities of rational space curves

  • Authors:

  • Xiaoran Shi;Xiaohong Jia;Ron Goldman

  • Affiliations:

  • Department of Mathematics, Harbin Institute of Technology, 150001, China and Beijing Computational Science Research Center, 100084, China and University of Science and Technology of China, Hefei, ...;KLMM, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080, China;Computer Science Department, Rice University, 6100 Main St., MS-132, Houston, TX 77005, USA

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2013

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Abstract

We provide a new technique to detect the singularities of rational space curves. Given a rational parametrization of a space curve, we first compute a @m-basis for the parametrization. From this @m-basis we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, we construct a new sparse resultant matrix for these three bivariate polynomials. We then compute the parameter values corresponding to the singularities by applying Gaussian elimination to this resultant matrix. Let @n"Q denote the multiplicity of the singular point Q, and let n be the degree of the curve. We find that when @?@n"Q=