Iqbal, R., Department of Computing Science, University of Newcastle upon Tyne
This thesis investigates, develops and implements algorithms for shape-preserving curve and surface design that aim to reflect the shape characteristics of the underlying geometry by achieving a visually pleasing interpolant to a set of data points in one or two dimensions. All considered algorithms are local and useful in computer graphics applications. The thesis begins with an introduction to existing methods which attempt to solve the shape-preserving curve interpolation problem using C1 cubic and quadratic splines. Next, a new generalized slope estimation method involving a parameter t, which is used to control the size of the estimated slope and, in tern, produces a more visually pleasing shape of the resulting curve, is proposed. Based on this slope generation formula, new automatic and interactive algorithms for constructing shape-preserving curves from C1 quadratic and cubic splines are developed and demonstrated on a number of data sets. The results of these numerical experiments are also presented. Finally, a method suggested by Roulier which generates C1 surfaces interpolating arbitrary sets of convex data on rectangular grids is considered in detail and modified to achieve more visually pleasing surfaces. Some numerical examples are given to demonstrate the performance of the method.