Title of Thesis

Data Visualization Using Spline Functions

Author(s)

MARIA HUSSAIN

Abstract
Accurate visualization of regular and scattered surface data requires that the surface characteristics or shape is preserved. This is desirable in most computer aided engineering applications, including; geometric modelling, sectional drawing, designing pipe systems in chemical plants, surgery; designing car bodies, ship hulls and airplanes; physical and chemical processes, geology, meteorology.
Three basic surface data shape characteristics, namely positivity, monotonicity and convexity are of general interest. For examples rainfall data is positive, the rate of dissemination of drugs in the blood is positive and monotone, data generated in an optimization problem may be convex.
Within a data visualization environment, a user is usually interested in graphically. This requires the use of interpolating schemes which themselves must possess certain characteristics like shape preservation, shape control, etc.
Many authors derived the constraints on derivatives to visualize the shape of data. These schemes fail to preserve the shape of data, when data are given with the derivatives at the data points. Some existing schemes are global, the disadvantages of these schemes: modification of data or constraints in one of the interval will affect the graphical display of the data over the whole domain. In the visualization of monotone scattered data, some existing schemes transform the scattered data in to the regular data. These schemes are
not feasible to industrial applications where bulk of data is under consideration.
The focus of this thesis is on the graphical display of regular and scattered surface data which possess positive, monotone and convex shape features. The aim is to develop the data visualization schemes that are local, computationally economical, visually pleasing that are applicable to both data and data with derivatives and above all provide automotive techniques for appropriate choice of parameters.
Data visualization schemes for regular data are developed using rational bi-cubic function and rational bi-cubic partially blended function to preserve positivity, monotonicity and convexity of surface data. Simple sufficient data dependent shape preserving constraints are derived in terms of the free parameters of rational bi-cubic and rational bi-cubic partially blended function.
The problem of data visualization of constrained data is also addressed when the data is lying above the plane and the interpolating surface is required to lie on the same side of the plane. Finally, data visualization schemes are developed for scattered data arranged over the triangular grid.

1

1.1 Motivation

1.2 Review of Literature
1.3 Outline of the Thesis

2.1 Introduction

2.2 Rational Cubic Function
2.3 Rational Bi-cubic Function
2.4 Positive Surface Data Interpolation
2.5 Constrained Surface Data Interpolation
2.6 Monotone Surface Data Interpolation
2.7 Convex Surface Data Interpolation

3.1 Introduction

3.2 Rational Cubic Function
3.3 Positive Curve Data Interpolation
3.4 Constrained Curve Data Interpolation
3.5 Monotone Curve Data Interpolation
3.6 Convex Curve Data Interpolation
3.7 Bi-cubic Partially Blended Rational Function
3.8 Positive Surface Data Interpolation
3.9 Constrained Surface Data Interpolation
3.10 Monotone Surface Data Interpolation
3.11 Convex Surface Data Interpolation

4.1 Introduction

4.2 C¹ Side-Vertex Method for Interpolation over a Triangle
4.3 Rational Cubic Function
4.4 Positive Scattered Data Interpolation
4.5 Monotone Scattered Data Interpolation

169