Graphics Reference
In-Depth Information
Figure 5.18.
Problem with sweeps.
Figure 5.19.
Generalized cylinder.
Sweeps sometimes become inputs to other representations. For example, in CADD
(a program developed by McDonnell Douglas) one can translate certain sweep repre-
sentations, such as translational and rotational ones, into boundary representations.
Related to sweeps is the multiple sweeping operation using quaternions described
in [HarA02]. There are also the generalized cylinders of Binford ([Binf71]). See Figure
5.19. Here the “sweeping” is parameterized. We shall now discuss a representation
scheme developed by J. Snyder at Caltech that is more general yet. It was the basis
for the GENMOD modeling system, which Snyder's topic [Snyd92] describes in great
detail.
Definition.
A
generative model
is a shape generated by a continuous transformation
of a shape called the
generator
.
Arbitrary transformations of the generator are allowed. There is no restriction as
to the dimension of the model. The general form of a parameterization S(u,v) for a
generative model which is a surface is
(
)
=
(
()
)
Suv
,
f
g
u v
,
,
(5.3)
where g : [a,b] Æ
R
3
is a curve in
R
3
and f :
R
3
¥
R
Æ
R
3
is an arbitrary function. One
of the simplest examples of this is where one sweeps a circle along a straight line to
get a cylinder. Specifically, let
[
Æ
Æ
(
3
g
:,
01
R
)
u
cos
2
p
u
, sin
2
p
u
,
0
be the standard parameterization of the unit circle. Define f by
(
)
=+
(
)
f
pp
,
v
00
,
,
v
.
