Graphics Reference
In-Depth Information
what computability might mean in the continuous rather than discrete setting. Finally,
Section 5.12 finishes the chapter with some comments on the status and inadequa-
cies in the current state of geometric modeling.
5.2
R-Sets and Regularized Set Operators
One of the terms that is used a lot in geometric modeling is the term “solid.” What
does it mean? It should be very general and include all the obvious objects. In par-
ticular, one would want it to include at the very least all linear polyhedral “solids.”
One also wants the set of solids to be closed under the natural set operations such as
union, intersection, and difference.
Intuitively, a solid is something that is truly three-dimensional and also
homo-
geneous
in the sense that, if we take a solid like the unit cube and stick a (one-
dimensional) segment onto it forming a set such as
X
=
[]
¥
[]
¥
[
»
(
[
) (
)
]
01
,
01
,
01
,
111 222
, ,
,
,
,
,
(5.1)
which is shown in Figure 5.2, then we do
not
want to call
X
a solid. A definition of a
solid needs to exclude the existence of such lower-dimensional parts.
Let
X
Õ
R
n
. Define the
regularization operator
r and the
regularization
of
Definition.
X
, r
X
, by
(
()
)
r
X
=
l
int
X
.
The set
X
is called a
regular set
or an
r-set
(in
R
n
) if
X
= r
X
, that is, the set is the
closure of its interior.
Note that the definitions depend on the dimension n of the Euclidean space under
consideration because the interior of a set does. For example, the unit square is an r-
set in
R
2
but not in
R
3
(Exercise 5.2.1). Note also that the set
X
in equation (5.1) is
not an r-set because
(
()
)
=
[]
¥
[]
¥
[
π
cl int
X
01
,
01
,
01
,
X
.
One can also show that
()
=
rr
XX
(5.2)
Figure 5.2.
A nonsolid.
