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

Sonia Pérez-Díaz and Li-Yong Shen
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Sonia Pérez-Díaz: Dpto de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain
Li-Yong Shen: School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Mathematics, 2021, vol. 9, issue 6, 1-21

Abstract: The ? -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 ? -bases is still developing, especially of surfaces. We study the ? -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 ? -basis, an inversion formula for a given proper parametrization P ( t ¯ ) is obtained. In addition, the degree of the rational map ? 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 ? -basis. Examples are given to illustrate the computational processes of the presented results.

Keywords: ?-basis; rational surfaces; inversion; improper; reparametrization; implicitization; resultant (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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