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

Sonia Pérez-Díaz,Li-Yong Shen

doi:10.3390/math9060640

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¯=(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.

Highlights

The study of representations of rational curves and surfaces is a fundamental task in computer aided geometric design (CAGD) and computer algebra
After we give the definition of μ-basis of a rational parametric surface defined parametrically by P ( t ), we find the inversion formula and the degree of the rational map that is induced by P while using the μ-basis
We study the μ-basis further for improper rational surfaces

Summary

Introduction

The study of representations of rational curves and surfaces is a fundamental task in computer aided geometric design (CAGD) and computer algebra. The algebraic equation, which is called implicit equation, is another important representation, and this is much better than the parametric expression in determining whether or not a point is on the curve or surface. For algebraic curves, is well-known that the existence of a proper reparametrization for a given improper rational parametrization is certified by Lüroth’s Theorem [28]. The μ-basis developing as a new algebraic tool can be used to obtain the parametric equation of a rational curve or a rational surface, in order to compute the implicit equation defining these varieties, and to study singularities and intersections [33]. An immediate consequence of the above theorems is that if p, q, r form a μ-basis if and only if p, q, r are a basis of syz(℘1 ( t ), ℘2 ( t ), ℘3 ( t ), ℘4 ( t ))

Inversion and Degree Using μ-Basis

On the Problem of the Reparametrization Using μ-Basis

Implicitization Using μ-Basis

Conclusions