Implicitization, Parametrization And Singularity Computation Of Surfaces Via μ-basis

In the field of Computer Aided Geometric Design (CAGD), there are two com-mon ways to represent curves and surfaces: parametric form and implicit form. Each of the two forms has its own advantages and disadvantages. For example, it is convenient to render and manipulate parametric surfaces. On the other hand, it is relatively easy to do inside-outside tests for implicit representations. If we hold the two forms at the same time, it is valuable for surface/surface intersection algorithms and other applications.In geometric modeling, a geometric model is generally represented in one form depending on applications. Hence the transformations between the parametric form and implicit form become important problems in CAGD and CG. The process of converting parametric form into implicit form is called implicitization, and the in-verse process is called parametrization. From classical algebraic geometry, any parametric form can be implicitized. However, the converse is not always true. Currently, there exist several typical implicitization methods including resultant methods, Grobner bases method. Wu's method. However, these methods have some defects on efficiency, generality and computing complexity respectively. The moving planes (surfaces) method and its generalization-μ-basis method, which were pro-posed by T.W.Sederberg and Falai Chen et.al., has good performance on the above aspects. From theμ-basis, we can easily recover the origin parametric curves (sur-faces). Furthermore, we can represent the implicit equations of curves (surfaces) in a more compact form. This implies that theμ-bases set up a connection between the parametric forms and the implicit forms of curves and surfaces.In this thesis, we focus on the implicitization and parametrization of lower degree rational surface based on moving planes andμ-basis theory. We also present a general implicitization framework for a general rational surface.In Chapter 2, we discuss the relations between moving planes and singularities of rational parametric surfaces, which is the foundation of singular locus computa-tion for the following two chapters. Moreover, an efficient algorithm is provided to compute the parametric loci of the self-intersection curves as well as their orders. The isolated singular points of the rational surface are also computed. In Chapter 3 and Chapter 4, we discuss the implicitization and parametriza-tion of quadratically parametrized surfaces, and methods to detect and compute the singular points are also presented. The key to connect implicit form and para-metric form is the moving planes with total degree one in s, t (called weakμ-basis). From the parametric form, we can get its weakμ-basis easily, from which the im-plicit form and singular points can be derived. Conversely, by finding the singular points from the implicit equation, we can get the weakμ-basis and its quadratically parametrization.In Chapter 5, we introduce a new type ofμ-basis and present a general frame for the implicitization of rational surfaces. Using blending functions, we can get moving quadrics or moving cubics from theμ-basis. These moving surfaces provide us a compact representation for the implicit equation. Further research problems are also discussed.