Computation Of The μ-bases Of Rational Curves And Surfaces And Its Application

Parametric form and implicit form are two common representations of curves and surfaces, and widely used in Computer Aided Geometry Design and Computer Graphics. These two representations show their advantages and disadvantages in different applications. In geometry modeling, a proper representation will be chosen according to certain application situations. Hence the transformations between the parametric form and implicit form, i.e., implicitization and parametrization become important problems in CAGD and CG. Currently, there exist several typical implicitization methods including resultant method, Grobner basis method, Wu's method, moving curve/surface method and μ-basis method. Among them, the moving curve/surface method and the μ-basis method are still developing. Especially, the μ-basis can not only be used in implicitization but also has some good properties and applications. From the μ-basis, we can easily recover the origin parametric curves/surfaces. On the other hand, we can also obtain the implicit equations of curves/surfaces in a more compact representation. This means that the μ-basis builds a connection between the parametric form and the implicit form of curves/surfaces.Since the μ-basis method is a newly developing method, there are many new problems to study. In the thesis, we focus on some problems and get some results, and also present some interesting ideas. Specially, we study the computation of the μ-basis and its applications in geometric modeling.First of all, in Chapter 2, we review some works on curves/surfaces, especially in implicitization and parametrization. In implicitization, we introduce the moving curve/surface method and the μ-basis method. It includes the definition, properties and some calculation algorithms of the μ-basis. The theories of the μ-basis of planar rational curves and rational ruled surfaces are reviewed and the corresponding algorithms are also presented.For general rational surfaces, the existence of their μ-basis has been proved, but the(?)e does not exist any algorithms to compute the μ-basis in a feasible approach. In Chapter 3, we reprove the existence of the μ-basis of general rational surfaces in polynomial matrix language and design a new μ-basis algorithm by using polynomial matrix operations. This algorithm can compute the μ-basis of any rational curves and surfaces. Especially in curves, it is more efficient than the present algorithm.It is obvious that space parametric curves can be implicitized by any typical methods, such as Gr(o|¨)bner basis method. But from the literature, we cannot find any systemic discuss on this topic. In Chapter 4, we study calculation and properties of the...