Jamil, Siti Jasmida
(2019)
Shape Preserving Interpolation Using
Rational Cubic Ball Triangular
Patches.
PhD thesis, Universiti Sains Malaysia.
Abstract
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.
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