The μ-basis of improper rational parametric surface and its application
AuthorsPérez Díaz, Sonia; Shen, Li-Yong
IdentifiersPermanent link (URI): http://hdl.handle.net/10017/49653
Agencia Estatal de Investigación
Pérez Díaz, S. & Shen, L.Y. 2021, "The μ-basis of improper rational parametric surface and its application", Mathematics, vol. 9, no. 6.
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-88796-P/ES/COMPUTACION SIMBOLICA: NUEVOS RETOS EN ALGEBRA Y GEOMETRIA Y SUS APLICACIONES/
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
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¯=(t1,t2) 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 ϕP 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.
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