Implicitization and parametrization of quadratic surfaces with one simple base point

  • Authors:

  • Xuhui Wang;Falai Chen;Jiansong Deng

  • Affiliations:

  • University of Science and Technology of China, Hefei, Anhui, China;University of Science and Technology of China, Hefei, Anhui, China;University of Science and Technology of China, Hefei, Anhui, China

  • Venue:
  • Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
  • Year:
  • 2008

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Abstract

This paper discusses implicitization and parametrization of quadratic surfaces with one simple base point. The key point to fulfill the conversion between the implicit and the parametric form is to compute three linearly independent moving planes which we call the weak u-basis of the quadratic surface. Beginning with the parametric form, it is easy to compute the weak u-basis, and then to find its implicit equation. Inversion formulas can also be obtained easily from the weak u-basis. For conversion from the implicit into the parametric form, we present a method based on the observation that there exists one self-intersection line on a quadratic surface with one base point. After computing the self-intersection line, we are able to derive the weak u-basis, from which the parametric equation can be easily obtained. A method is also presented to compute the self-intersection line of a quadratic surface with one base point.