دانلود رایگان مقاله مجتمع μ-پایگاه برای سطوح چهار تایی واقعی

1. Introduction

μ-Bases are a powerful tool for analyzing the algebraic and geometric properties of rational curves and surfaces. For example, μ-bases for rational planar curves can be used to easily retrieve both the implicit and parametric equations of these curves as well as to locate and analyze their singular points (Chen and Wang, 2003a; Chen et al., 2008; Jia and Goldman, 2009, 2012). Straightforward, efficient techniques are known for computing μ-bases for rational curves of arbitrary degree in arbitrary dimensions (Song and Goldman, 2009), but computing μ-bases for rational surfaces of arbitrary degree in just 3-dimensions is, in general, notoriously difficult (Chen et al., 2005; Deng et al., 2005). Currently the only exception to this seemingly intractable problem for high degree surfaces is constructing μ-bases for rational ruled surfaces from parametrizations of bidegree (1,n), which essentially can be reduced to computing μ-bases for rational curves (Chen and Wang, 2003b; Chen et al., 2001). In contrast, for surfaces in 3-dimensions with rational quadratic parametrizations there do exist straightforward techniques for finding a μ-basis. A method for computing μ-bases for quadric surfaces with rational quadratic representations is presented in Chen et al. (2007). The construction involves solving two systems of linear equations, each with 12 unknowns, to generate three moving planes that follow the surface. The implicit, parametric, and inversion equations for the surface can all be retrieved from these three moving planes. Analogous techniques for finding μ-bases for cubic and quartic surfaces with rational quadratic parametrizations are presented in Wang et al. (2008), Wang and Chen (2012).

5. Conclusions

The use of complex parameters and complex moving planes that follow a rational surface works nicely for finding complex analogues of μ-bases for non-ruled quadric surfaces. But this method is not a panacea. We have seen that the same method, without complex parameters, can be used to find the implicit equation of a ruled quadric surface with a rational quadratic parametrization, but in general for ruled quadric surfaces this method fails to retrieve either the parametric equations or the inversion formulas. Moreover, different parametric representations for the same algebraic surface may have different syzygy modules. Thus the method can fail even for higher degree parametrizations of non-ruled quadric surfaces. For example, there is a biquadratic parametrization of the sphere given by x(s,t) = (1 − s 2)(1 − t 2), y(s,t) = 2s(1 − t 2), z(s,t) = 2t(1 + s 2), w(s,t) = (1 + s 2)(1 + t 2). Since the implicit degree of the sphere is two, this parametrization has six base points, four of which are located at t = ±1, s = ±i. Yet there are no moving planes of the form (Ax + B y + C z + D w)(s ± i) + (Ex + F y + G z + H w)(t ± 1) that follow this parametrization because we have only 8 free parameters, but there are 15 − 4 = 11 constraints. Thus the method does not work, in general, for surfaces of bidegree (2, 2), even when there are lots of base points.