Graphics Reference
In-Depth Information
(7) transforming shapes by scaling, mirroring, rotating, translating, . . .
In addition to the geometric operations, commercial modeling systems must also
have the ability to annotate the geometry so that users can produce standard mechan-
ical drawings. As a matter of fact, dimensioning and tolerancing is not an after-
thought, but an important aspect of the total design and manufacturing process.
Dimensions specify the “nominal” or perfect geometry of an object. Since one can
never create the perfect object, one needs to be able to specify tolerances within which
it would be considered acceptable. Other annotations relate to the type of material of
an object and how it is to be rendered - in what color, etc. The representation schemes
described in this chapter dealt with dimensions. The representation problem for tol-
erances is much harder. Coming up with a rigorous theoretical foundation for what
humans have been doing manually for hundreds of years has been a nontrivial task
and many problems remain as one moves towards the goal of automating annota-
tions. See [Just92] for a brief overview of this topic.
Another property that can distinguish modelers is whether they use exact or
faceted representations of objects in computations. Many modelers use a faceted rep-
resentation for display purposes because the display hardware has been optimized to
deal with that. The display itself is not the issue here, but rather, whether an algo-
rithm, such as one that determines if two objects intersect, uses a faceted or exact
representation of these objects. Finding the intersection of linear spaces is much
easier than doing this for curved spaces and so many modelers did this in the 1980s.
There are potentially serious problems with this however. To give a two-dimensional
example, consider what happens when slightly overlapping circles are approximated
by polygons. If they are rotated, an intersection algorithm would sometimes find an
intersection and sometimes not, like in Figure 5.52. An incorrect determination of this
type could cause problems if passed on to another operation.
This chapter has presented an overview of the evolution of geometric modeling.
Although we have come a long way, the current state of the field is far from satisfac-
tory. Attempts have been made to develop theoretical foundations so that one can talk
about issues in a rigorous way, but by in large the field is still advancing in an ad hoc
way. Some specific defects are:
(1) R-sets may be inadequate. One may want to be able to represent non-
manifold solids with dangling edges or faces. Simply enlarging the domain
would still leave unanswered the question of which operations to allow and
what they would preserve. See [GHSV93].
(2) In practice the definition of a representation scheme r is rather ad hoc. Usually
only r
-1
is well-defined and r itself is not. It is hard to compare different rep-
resentation schemes. For example, when converting from a boundary to a CSG
Figure 5.52.
An intersection problem with faceted
circles.
