Graphics Reference
In-Depth Information
Definition.
The
generative modeling representation
consists of pairs (
X
,F), where F
is a parameterization of the generative model
X
of the form shown in equation (5.4).
The driving force behind GENMOD was correcting some perceived deficiencies in
the geometric modeling systems of that time and some key defining points listed by
[Snyd92] for the generative modeling approach as implemented in GENMOD are:
(1) The representation is a generalization of the sweep representation.
(2) Shapes are specified procedurally.
(3) Specifying a shape involves combining lower-dimensional shapes into higher-
dimensional ones.
(4) An interactive shape description language allows low- and high-level opera-
tors on parametric functions.
(5) It is closed, that is, the outputs to operations can be inputs to operations (like
CSG).
(6) It allows parameterized shapes whose parameters a user can change.
(7) It supports powerful high-level operators and functions, such as
reparameterizing a curve by arc length,
computing the volume of a shape enclosed by surface patches, and
computing distances between shapes.
These operations are closed and free of approximation error.
(8) It supports deformation operators, CSG, and implicitly defined shapes.
(9) One has the ability to control the error in the representation.
A large variety of symbolic operators on the parameterizations and their coordi-
nates help the user define generative models, such as vector and matrix operations,
differentiation (partial derivatives), integration, concatenation, and constraint opera-
tors. Since parameterizations can be thought of as vector fields, another useful oper-
ator is one that solves ordinary differential equations. GENMOD had a language in
which a user could define models using the various operators.
Now, models will have to be displayed. By converting to polygonal meshes and ad
hoc error control, the interactive rendering of generative models becomes feasible.
One can specify the subdivisions in two ways: uniform in domain or adaptive sam-
pling. More realistic images can be obtained at the expense of speed.
For accuracy, GENMOD used interval analysis. Interval analysis (see Chapter 18)
is an attempt to make numeric computations on a computer more robust and has its
advantages and disadvantages. Snyder argued for its use in geometric modeling and
described various applications to computing nonintersecting boundaries of offset
curves and surfaces, approximating implicitly defined curves and surfaces, and
trimmed surfaces and CSG operations on them.
In summary, three more advantages used by Snyder to justify the generative
modeling approach are:
(1) The representation handles all dimensions, is high-level, and extensible.
(2) Using a high-level interpreted language, the mathematically knowledgeable
user can easily build a library of useful shapes.
(3) An adequate number of robust tools for rendering and manipulating genera-
tive models exist.
