Positivity-preserving Interpolation Of Positive Data By Rational Cubics And Their Approximation Properties

Rational spline is the natural generalization of polynomial spline.However,the space of rational spline is more complicated,so the theories are not as perfect as those of polynomial case.Some questions should be further studied.With the prevalence and progressively broad application of computer technology,shape-preserving interpolation has become a focus of study in Computer Aided Geometric Design and Computer Graphics in the world and it is very important in Numerial Control Machine.This dissertation studies positivity-preserving interpolation.We focus on C~1 rational spline with quadric denominator.Moreover,we consider an interpolating rational bicubic form for the data arranged over a rectangular grid.The paper is arranged as follows.In Chapter 1,the author makes a brief introduction of the development of Computational Geometry.The development of spline interpolation,especially the rational cubic spline interpolation,is introduced.In Chapter 2,one kind of rational cubic spline interpolation with quadric denominator has been proposed.Furthermore,the approximation property is analyzed under the function interpolated is c~2.Simple sufficient condition is developed on the free parameter in the description of rational function to visualize the positive data the form of the positive curve.In Chapter 3,the author analyzes the properties of the rational cubic spline interpolation based on arithmetic average difference quotient under two conditions where the function interpolated is C~1 or C~2.In Chapter 4,the method which is proposed in Chapter 2 is extended to an interpolating rational bicubic form for the data arranged over a rectangular grid.Furthermore, the approximation property is analyzed under the function interpolated in C~2.