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(1) horizontal line A of length a
(2) line B perpendicular to line A of length b
(3) line D makes signed angle a with line B
(4) circular arc C tangent to lines A and D
It would then be up to the system to check if this description gives rise to a unique
consistent model for particular values of a and b. The basic design process in such
a modeling system is then that the user describes a list of geometric primitives to
be used and the geometric relationships between them symbolically without any
numbers. After the user inputs the actual geometric constraints, the system creates
an actual instance of the object if possible. The user can subsequently input new data
and get new instances of the object. Note that the “parametric” models considered
here are higher-level constructs than those in the generative representation discussed
in Section 5.3.5.
Although the terms “parametric” and “variational” are often used interchangeably,
there is a subtle distinction between what are called parametric and variational
methods. Parametric methods solve constraints by replacing symbolic variables by
values that are computed sequentially from previously computed variables. Varia-
tional methods use equations to represent constraints and solve them simultaneously.
The difference is captured by the difference between defining a variable via a formula
or implicitly. For more on parametric and variational modeling see [ShaM95]. Some
sample papers on constraint-based modeling with additional references are [LiHS02]
and [Podg02].
An approach to geometric design based on functional programming that extends
variational modeling is described in [PaPV95]. The authors discuss a high-level func-
tional programming language (a language that manipulates functions) with the under-
lying geometric objects represented in a hierarchical manner much like in CSG.
Elementary polyhedra are stored as inequalities and the basic Boolean set operations
are supported. The language is such that all syntactically correct objects are valid. It
is argued that the power of this functional approach is that it lets the user naturally
generate new models from old ones and is similar to generative modeling in this
respect.
Parametric and variational modeling is a start toward facilitating geometric
design, but it still only deals with individual geometric primitives with no grouping
capabilities and lacks a vision of the whole modeling process. Consider a manufac-
turing company. Its ability to deal with the design, planning, and manufacturing
process in an integrated way is clearly of practical importance. To do this one needs
to model the whole process. However, the models used by a designer should be allowed
to be different from those used by the person manufacturing the product about which
the designer may know little. Both may evolve over time and one only needs a way to
map from one to the other. Feature modeling seems like a promising approach to an
integrated solution. Again, the problem with the type of modelers we have been dis-
cussing up to now is that they dealt solely with the geometry of objects and ignored
many of the other issues such as process planning, assembly planning, and inspec-
tion planning. Even in the case of just the geometry they were not totally adequate
since they tended to be low-level and did not make it easy for a designer to make
changes, although parametric modeling helped.
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