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where one tries to establish in advance what a user may want to do. Much more work
needs to be done in this area. The then-current state of feature-based design and key
problems are discussed by Mäntylä et al. in [MäNS96]. Developing expert systems in
the geometric context is much harder than for the traditional areas where expert
systems have been successful. Ways of using a blend of artificial intelligence and com-
putational geometry algorithms to help here are described by Requicha in [Requ96]).
Amato's ([Amat96]) view of geometric modeling was that while a lot has been done
with respect to
building
models, much more work remains to be done when it comes
to
analyzing
and
manipulating
models. She specifically addresses the problem of
virtual prototyping in industrial design. There are many more stages to product devel-
opment than just modeling and one would like to automate them also. Two of the
problems she mentions are:
Design for Maintenance.
After individual parts of a larger object have been
designed and assembled, one would like an understanding of how they fit “together.”
Can a part be removed without removing other parts? Is there room enough to insert
a tool and for a mechanic to do this? Amato points out that the Boeing 777 was
completely designed in a CAD/CAM system, but the maintenance issues were studied
with physical models. A problem such as part removal is really a special case of the
difficult motion planning problem.
Assembly Sequencing.
If an object is made up of parts, what sequence of steps
would allow one to put the whole object together? This is related to the part removal
problem.
By modeling the entire prototyping process, one can create manuals and also
virtual environments for training. Computational geometry is a major component in
a solution because many of the problems that have to be solved belong to that field.
From now on we shall use the term “geometrically intelligent modeling system”
to mean a system that understand geometric invariants associated to spaces and can
answer questions such as whether or not two spaces are homeomorphic or isometric.
We are talking about systems that are knowledgeable about the topology
and
differ-
ential geometry of an object. Current commercial systems (basically those outside
university research departments) only have a superficial knowledge of the geometry
of objects. Most of what they know is
local
information.
For a modeling system to be geometrically intelligent, it will minimally have to
maintain whatever is necessary to compute the invariants described in Chapters 6-9
in [AgoM04]. These invariants enable us to distinguish between surfaces. Although
they are insufficient for classifying spaces in higher dimensions, they are at least
a start towards that goal. The following two steps are necessary to carry out this
program:
(1) We need to maintain an adequate cell structure for all of its objects, one suit-
able for computing at least the standard algebraic topology invariants such as
the homology groups.
(2) We need to maintain the appropriate differentiable structure to compute dif-
ferentiable geometry invariants. In the discrete case one only needs the attach-
ing maps of the cells.
