Shape Preserving Interpolation And Shape Control Of Rational Functions

This thesis is aimed to investigate C~1 monotonicity preserving piecewise rational cubic interpolation; C~1 convexity preserving of piecewise rational cubic interpolation; shape control of piecewise rational cubic interpolating spline using both function values and derivatives of the function being interpolated as the interpolation data, and shape control of piecewise rational cubic interpolating spline only based on function values of the function being interpolated as the interpolation data. The interpolating function contains some adjustable parameters. The interpolation functions not only have simple mathematical representation, but can be used for the modification of local curves by selecting suitable parameters under the condition that the interpolating data are not changed. A series of new results are obtained. Many related results reported in the literature are extended or improved.In Chapter 1, the background of shape preserving interpolation, shape control of rational cubic functions and the theoretical and practical significance of main works are introduced.In Chapter 2, the monotonicity preserving piecewise rational cubic interpolation functions with linear denominator or quadratic denominators or cubic denominators are constructed, the interpolating functions are C~1 continuous. Because three kinds of the interpolating function expressions have adjustable factors, the interpolation curves have more flexibility.In Chapter 3, piecewise rational cubic interpolation function with linear denominator or quadratic denominators or cubic denominators are constructed, the interpolating functions are C~1 continuous. Two methods are presented for controlling the convexity of interpolating curves. The conditions for the interpolating curves to be convex in the interpolating intervals. So that for the given data the shape of the interpolating curve can be modified by selecting suitable parameters.In Chapter 4, a rational cubic interpolating spline with cubic denominators is constructed using both function values and derivatives of the function being interpolated as the interpolation data. The sufficient conditions for the interpolating curves to be above, below or between the given broken lines or piecewise quadratic curves are derived.In Chapter 5, a rational cubic interpolating spline based on function values and with quadratic denominators is constructed. The sufficient conditions for the interpolating curves to be above, below or between the given broken lines or piecewise quadratic curves and the sufficient and necessary conditions for the interpolating curves to be above, below or between the given broken lines are derived. Next, a rational cubic interpolating spline based on function values and with cubic denominators is constructed. The sufficient conditions for the interpolating curves to be above, below or between the given broken lines or piecewise quadratic curves are derived.