MATHEMATICAL ALGORITHMS OF CAGD
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Date
1991-03
Authors
HJ. AWANG, MOHD. NAIN
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Abstract
CAGD is the abbreviation for "Computer-Aided Geometric
Design". This is a study of mathematical represent at ion of curves
and .. surfaces which is suitable for computer graphics .display and
rendering. It has wide applications in industries such as car,
aircraft and ship building. It also cont~ibutes to the advancement
of the research in the area of image analysis and pattern
recognition. The area of research which began in early 1960 has
become one of the active research area among mathematicians,
engineers and also computer scientists.
In CAGD, curves and surfaces are defined parametrically and
piecewice manner. Initially, non-rational polynomials were employed,
but since early 1980 more research work were focussed in the
rational form. In fact, in some of the aircraft industries, the use
of application packages such as CADAM which uses rational B-splines
for the generation of curves and surfaces were considered. In order
to define rational curves and surfaces, we have to make use of
homogeneous coordinates, namely by adding one more dimension to the
space involved. The rational form reduces to the non-rational by
using appropriate projection.
In CAGD, efficient algorithm for generating curves and
surfaces are required.
The aim of this thesis is to consider various aspects for the
application of algorithms in the construction of curves and
surfaces. The study wi 11 consider the Bezier, B-splines and the
homogeneous polynomial Splines methods.
We start in Chapter 1 by reviewing the Bezier curves and
surfaces, and discuss some of their properties. Then in Chapter 2,
we ·wi 11 consider the construct ion of curves and surfaces by using
univariate splines. Subdivision algorithm for generating
univariate spline curves and surfaces will be considered in Chapter
3. In Chapter 4, we also construct B-splines over a regular
triangulation by convolutions, and these will form box spline.
In Chapter 5, rational Bezier and rational B-spline curves and
surfaces are considered. The recursive algorithms are used to
construct these curves and surfaces. Moreover, we show how knot
vectors and weight functions in the definition of the rational form
wi 11 provide some means of control over the shape of Bezier and
B-spline curves and surfaces.
Finally, in Chapter 6, we construct functions which are
piecewise homogeneous polynomials in the positive octant in three
dimensions. By using a linear transformation followed by a
projection on suitable planes, we obtain piecewise polynomial
functions of two variables on a mesh formed by the three pencils of
lines. The vertices of these pencils may be finite or one or two
may be infinite. As a limiting case all three vertices become
infinite and one recovers polynomial box splines on three direction
mesh.
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MATHEMATICAL ALGORITHMS OF CAGD