Research On Some Topics In Nonlinear Scientific Computing

This paper mainly makes some research on the existence of rational interpolation and methods of curves and surfaces modeling whose problems in trigonometric polynomial. It includes the existence of univariate and bivariate rational interpolation, the properties and applications of B-L curves and surfaces and the applications of the uniform B-L spline curves and surfaces with parameters.In the first part of this paper, we review the background of the study of the existence of rational interpolation, the classification and characteristics of curve and surface modeling, and the history of the parameter curves and surfaces modeling in CAGD, and briefly introduce Bézier, B-spline and non-polynomial curves and surfaces modeling, which include L-splines, helix splines, splines in tension and C-curves etc.In the second part of this paper, we firstly give a method to test the existence of a class of bivariate rational interpolation based on the research of Prof. Zhu Xiao-lin. The method is convenient for calculating and has some inheritance. Some numerical examples are given to illustrate the results. Secondly, by analyzing the characteristics of Bézier curves and B-spline curves, we construct some trigonometric polynomial curves in the space of trigonometric functions, which possess simple expressions and the characteristics similar to Bézier curves and B-spline curves. We call them simulant B-L curves and B-L-spline curves. B-L curves not only inherit the advantages of Bézier curves and B-spline curves, but can also be used to represent straight lines and some transcendental curves precisely, such as circular arc and ellipse etc. The B-L curves and B-L-spline curves can be converted to rational curves easily and therefore can take part in the current system of geometry modeling. Thirdly, we make a generation of the B-L curves with lower-order and present thier explicit expressions. Finally, we study the uniform B-L spline curves and surfaces with some parameters. All the above B-L curves and B-L spline curves are generated to their corresponding tensor product surfaces.