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The μ -Basis of Improper Rational Parametric Surface and Its Application

Author

Listed:
  • Sonia Pérez-Díaz

    (Dpto de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain
    These authors contributed equally to this work.)

  • Li-Yong Shen

    (School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
    Key Laboratory of Big Data Mining and Knowledge Management, CAS, Beijing 100190, China
    These authors contributed equally to this work.)

Abstract

The μ -basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ -bases is still developing, especially of surfaces. We study the μ -basis of a rational surface V defined parametrically by P ( t ¯ ) , t ¯ = ( t 1 , t 2 ) not being necessarily proper (or invertible). For applications using the μ -basis, an inversion formula for a given proper parametrization P ( t ¯ ) is obtained. In addition, the degree of the rational map ϕ P associated with any P ( t ¯ ) is computed. If P ( t ¯ ) is improper, we give some partial results in finding a proper reparametrization of V . Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ -basis. Examples are given to illustrate the computational processes of the presented results.

Suggested Citation

  • Sonia Pérez-Díaz & Li-Yong Shen, 2021. "The μ -Basis of Improper Rational Parametric Surface and Its Application," Mathematics, MDPI, vol. 9(6), pages 1-21, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:640-:d:519013
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