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, visuallyv
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.