Research On Bivariate Diagonal Vector Valued Rational Interpolation And Distribution Of Singularities And Inflection Points On Curves

This thesis mainly makes some research on algorithms for bivariate diagonal vector valued rational interpolation and the inflection and singular points on FB-spline curves. It includes two kinds of algorithms for the bivariate vector valued rational interpolation over leading diagonal and sub-diagonal respectively, the matrix algorithm for computing bivariate diagonal vector valued rational interpolation with preassigned poles, distribution of singularities, inflection points, loops and cusps on FB-spline curves.In the first part of this thesis, we review the basic theory and method of the vector valued rational interpolation, the development of curve and surface modeling, the development of analysis of inflection and singular points on some kinds of curves, and briefly introduce the algorithm for the bivariate thiele-type vector valued branched continued fraction rational interpolation, the algorithm for bivariate vector valued rational interpolation over rectangle grids which is lack of datum, the definition of C-B spline curves, H-B spline curves and FB-spline curves, analysis of inflection and singular points on planar C-Bézier curves, analysis of singularities on rational Bézier curves.In the second part of this thesis, we firstly give two methods for computing bivariate diagonal vector valued rational interpolation, one is computing b_i,j directly, another is a kind of matrix method which is given by defining a special elementary operation in the sense of Samelson inverse. In addition we constructs a matrix algorithm for computing bivariate diagonal vector valued rational interpolation with preassigned poles, then some examples are given to illustrate the validity of the above algorithms. We discusses singularities, inflection points and convexity of FB-spline curves in terms of their control polygons, and gives the necessary and sufficient conditions for testing when FB-spline curves have one or two inflection points, or a cusp, or a loop, or none of the above points by the envelope theories and topological mapping method. At last, all kinds of distribution of singularities, inflection points, loops and cusps on FB-spline curves onλμ-plane are illustrated.