Approximate μ-Bases of Rational Curves and Surfaces

  • Liyong Shen
  • Falai Chen
  • Bert Jüttler
  • Jiansong Deng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4077)


The μ-bases of rational curves and surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of curves and surfaces. However, exact μ-bases may have high degree with complicated rational coefficients and are often hard to compute (especially for surfaces), and sometimes they are not easy to use in geometric modeling and processing applications. In this paper, we introduce approximate μ-bases for rational curves and surfaces, and present an algorithm to compute approximate μ-bases. The algorithm amounts to solving a generalized eigenvalue problem and some quadratic programming problems with linear constraints. As applications, approximate implicitization and degree reduction of rational curves and surfaces with approximate μ-bases are discussed. Both the parametric equations and the implicit equations of the approximate curves/surfaces are easily obtained by using the approximate μ-bases. As indicated by the examples, the proposed algorithm may be a useful alternative to other methods for approximate implicitization.


approximate μ-bases approximate implicitization 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Liyong Shen
    • 1
    • 3
  • Falai Chen
    • 1
  • Bert Jüttler
    • 2
  • Jiansong Deng
    • 1
  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaChina
  2. 2.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria
  3. 3.KLMM, Institute of Systems Science, AMSSChinese Academy of Sciences 

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