Shape Preserving Interpolation Using Rational Cubic Ball Triangular Patches - Jamil, Siti Jasmida (2019)
Shape preserving interpolation is an important area for graphical presentation of scattered data where it is most desired in computer graphics, computer aided manufacturing, computer aided geometric design, geometric modeling, geology, meteorology, as well as in physical and chemical process. In many interpolation problems, shape characteristics of the surface data commonly considered are positivity, monotonicity and convexity. Thus, the focus of this thesis is on the graphical displays of triangular surfaces of scattered data which possess positive, monotone and convex shape features, respectively. Shape preserving schemes will be displayed for triangular patches using rational cubic Ball function with free shape parameters (weights function). It will be shown that the proposed scheme is visually pleasing when appropriate parameters are chosen. Firstly, for each data set in two dimensional (2D) region (x,y) is divided into triangular elements using Delaunay triangulation method. The interpolating surface of scattered data is a convex combination of three rational cubic Ball triangular patches with the same set of boundary Ball ordinates. Conditions to obtain positivity, monotonicity and convexity preserving surfaces, respectively, are derived on the Ball ordinates with free parameters in order to preserve the inherited shape characteristics of the underlying data. Finally, a relationship between rational Bézier and rational Ball bases will be shown using conversion formulae.