Implicitization and parametrization of quadratic surfaces with one simple base point

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.

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

Additional Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

General Terms:
Algorithms

Keywords:
implicitization, moving plane, parametrization, weak u-basis