CURVES AND SURFACES FOR DATA VISUALIZATION USING SPLINES

Abstract

This thesis deals with the visualization of regular and irregular (scattered) data using spline curves and surfaces. For the spline curves, a 1 C rational cubic spline is proposed and developed with two free parameters together the error analysis discussed. The proposed rational spline is used to develop three new curve schemes to visualize the shaped data of the positive, monotone and convex data. Algorithms have been developed for the three shape preserving curves schemes. These curve schemes are practically demonstrated for various data in literature. For the visualization of regular grid data, the proposed curve interpolants are extended to their surface counter parts (rational bi-cubic functions) over the rectangular grid. Each boundary curve of the rectangular grid is constructed by the rational cubic functions having two free parameters in its description. This work has been further extended to device three new surface schemes for the visualization of shaped data by imposing data dependent constraints on free parameters; first scheme for the visualization of positive data, second for the visualization of monotone data and third for the visualization of convex data. The algorithms have been designed, for each of surface schemes, to efficiently compute the proposed shape preserving surfaces. The proposed surface schemes are practically demonstrated to shape preserving data. The degree of smoothness attained is 1,1 C . Lastly three schemes are introduced for the visualization of scattered data in the view of positive, monotone and convex surfaces. The given scattered data is triangulated over the domain and piecewise rational cubic function is used for the interpolation of boundary and the radial curves. The final visualized surface is the convex combination of boundary and the radial curves facilitating twelve parameters in each triangular patch. The data dependent constraints are derived on six of the free parameters for visualization of positive, monotone and convex shape of scattered data, while remaining six parameters are free for shape modification. These proposed schemes are local, computationally economical, do not constrain step length, and are equally applicable to data with derivatives and without derivatives.