Applications Of μ-bases—Computing The Singularities Of Rational Space Curves And Implicitization Of Rational Surfaces

The μ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singularities of rational planar curves/ruled surraces.In this thesis, based on the existing research results, we present more appli-cations of μ-bases in rational space curves and surfaces of revolution. For rational curves, we apply μ-bases to compute the singularities of rational space curves, and based on a new constructing bihomogeneous resultant matrix to analyze the mul-tiplicities of all singular points. For rational surfaces, we construct a μ-basis for surface of revolution, and a Sylvester-type matrix whose determinant gives the im-plicit equation of surface of revolution. And we extend the result to more general surfaces which has a pair of orthogonal directrices.In Chapter1, we first give a brief history of Computer Aided Geometric de-sign. We mention parametrization and implicitization are two important problems in CAGD, and summarize recent developments of μ-basis of rational curves and surfaces and its applications.In Chapter2, we provide some preliminary knowledge that we will use in the following chapters, including the definitions and known applications of μ-bases for rational planar curves, rational space curves and rational surfaces.In Chapter3, we discuss the singularities of rational space curves. Two methods are provided to compute the singularities of arbitrary degree curves. These methods are a generalization of the paper (Chen, Wang and Liu, Computing singular points of plane rational curves. Journal of Symbolic Computation43,92-117,2008), which are based on the μ-basis of the rational space curve and on random technique. The comparison between our two methods and a generalized resultants method are provided. Examples are provided to illustrate the effectiveness of our methods.In Chapter4, we provide a different technique to detect the singularities of rational space curves. From a μ-basis for a space curve, we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, a new sparse resultant matrix is constructed. We can get the information of the singularities through this resultant matrix by Gaussian eliminations.In Chapter5, we focus on μ-bases of surfaces of revolution. We compute a μ-basis for the surface of revolution from a μ-basis of its directrix and a rational parametrization of the circle. Then we construct a sparse Sylvester style resultant matrix and a Bezout style resultant matrix for three bivariate polynomials of bide-grees (1,μ),(1, n-μ),(2,0) which are derived from the μ-basis. Both determinants provide compact representations for the implicit equation of a rational surface of revolution.In Chapter6, we extend the result in Chapter5to surfaces generated by two orthogonal directrices. We compute a μ-basis from the μ-basis of its directrices and construct a Sylvester style resultant matrix whose determinant gives the implicit equation of the surface. But the constructing of the Bezout style resultant is hard work, which is an open problem now.