ylfink@water.UUCP (06/09/87)
DEPARTMENT OF COMPUTER SCIENCE
UNIVERSITY OF WATERLOO
SEMINAR ACTIVITIES
COMPUTER GRAPHICS SEMINAR
- Friday, June 12, 1987
Dr. Ron Goldman of Control Data Corporation will speak
on ``Future Directions in Geometric Modeling: A
Preview of Coming Attractions''.
TIME: 3:30 PM
ROOM: MC 6091A
ABSTRACT
Geometric modeling is a rapidly expanding field. Many
new, exciting, and powerful techniques are currently
emerging from industrial and academic research
laboratories. These methods will enable mechanical
engineers to model and simulate piece parts and
assemblies more effectively and efficiently than ever
before. In this talk we shall survey seven areas of
research which we believe will have a profound effect
on the future of mechanical engineering and geometric
modeling.
1. Freedom Deformations of Solid Models
Traditional solid modeling systems are based on
very simple and very rigid geometry, usually just
planar and quadric surfaces. Yet to model
realistic mechanical parts, freeform curves and
surfaces are clearly required. Freeform
deformation of solid models combines freeform
design with traditional solid modeling - e.g.
quadric surface - techniques. The method is easy
to program and simple to use. It permits the
implementation of freeform design on top of already
existing solid modeling systems. Thus it is bound
to have a profound effect on future computer models
of mechanical parts and assemblies.
2. Triangular Bezier Patches
Today it is common practice to model freeform
surfaces with rectangular patches. Yet there are
many parts which do not have a rectangular topology
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and so do not lend themselves well to rectangular
patches. Also if the data is scattered rather than
gridded, rectangular surfaces are not really
natural. Triangular surfaces are mathematically
simpler, lend themselves better to arbitrarily
topologies, and can be integrated easily with
standard rectangular patches. The mathematics of
these surfaces is now well enough understood so
that these patches will soon become standard tools
in surface design systems.
3. Analytic Surfaces
Parametric polynomial patches are the traditional
surface design tools of the computer graphics
community. Parameterization makes them
particularly easy to display and so they were
naturally the first freedom surfaces to be
exploited in computer graphics. However analytic
surfaces have many advantages over the standard
parametric patches. They include parametric
patches as a subclass, but are generally of lower
degree. Their low degree can be exploited to
produce faster more robust algorithms, and their
analytic form can be used to generate very nice
blending techniques for solid modeling systems.
New research now reveals that they can be used
quite readily for customary freeform design.
4. Geometric Continuity
Parametric continuity is generally unnecessarily
restrictive for geometric modeling. The mechanical
designer often knows nothing of the underlying
parameterization of his surface and cares only for
the visual or geometric smoothness of his patches.
Nu-splines and beta-splines exploit this freedom
from parameterization to introduce new parameters
called shape parameters which allow the designer to
refine the shape of a patch without altering the
control points. This gives the engineer finer
control over the ultimate shape of the design.
5. Urn Models and Blending Functions
Lagrange polynomials, Bezier curves, and B-splines
are among the standard weapons in the arsenal of
computer aided geometric design. Are these
techniques related in any way? How can we create
new techniques? Is there any mathematical unity to
this subject? Strangely enough, it appears that
standard discrete probability theory provides an
answer to many of these questions in approximation
theory and computer aided geometric design. We
shall show how to use simple stochastic models to
generate blending functions for computer aided
geometric design and in the process unify,
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generalize, and simplify many well known geometric
results.
6. A Multiprocessor Architecture for Curve and Surface
Design
One ultimate goal of computer graphics is the real
time manipulation of freeform curves and surfaces.
Most of the techniques now being used to create
freeform polynomial curves and surfaces have a
simple recursive form (see the section on urn
models and blending functions). This recursion can
be implemented in parallel hardware which can be
built from simple off the shelf processors. This
multiprocessor system can output one point per
clock cycle or roughly 2 million points per second.
Thus the real time manipulation of polynomial
curves and surfaces is well within reach of current
technology.
7. Feature Based Design
The language of geometric modeling is not the
language of the mechanical engineer. Geometric
modeling systems typically communicate in terms of
solids such as boxes, cylinders, spheres, or cones;
the mechanical engineer traditionally talks in
terms of features such as slots, holes, pockets,
and fillets. To bring geometric modeling closer to
the engineer, we need to design modeling systems in
which the designer can speak and think in terms
which are meaningful to the engineer rather than in
terms which are meaningful to the system. That is,
we must allow the engineer to design with concrete
features rather than abstract geometry. Feature
based design also has several additional benefits.
It permits design for manufacture; it can
incorporate dimensions and tolerances; and it leads
itself well to automatic classification of parts
and assemblies. Thus design with concrete features
rather than with abstract geometry will be the
preferred method in the modeling systems of the
future.
To summarize: new design paradigms are now emerging
which challenge the traditional ways in which we
perform geometric modeling. These techniques will
in some cases enhance and in other cases supplant
the way we currently perform computer aided
geometric design. The challenge before us now is
to incorporate these new methods into practical
industrial modeling systems for the working
mechanical engineer.
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June 9, 1987