Graphics Reference
In-Depth Information
Figure 5.4.
Quasi-disjoint sets.
Definition.
Define the
regularized set operators
»*, «*, -*, c*, and D* by
»=
( )
«=
( )
-= -
XY XY
XY XY
XY XY
YY
XYXY YX
*
r
r
r
,
*
,
(
)
*
,
=
()
=-
c
*
r c
,
and
(
)
»-
(
)
D
*
*
*
*
,
where c and D are the complement and symmetric difference operators, respectively.
5.2.1
Theorem
(1) The regularized set operators take r-sets into r-sets. Furthermore, there are algo-
rithms that perform these operations.
(2) The class of regular semialgebraic or semianalytic sets is closed under regularized
set operations.
Proof.
For (1) see [Tilo80] or [Mort85]. For (2) see [Hiro74].
Even though r-sets are quite general, they have their limitations.
(1) Although they have attractive features from a mathematical point of view, they
are complicated to deal with computationally.
(2) One may want to deal with nonsolids like in Figure 5.2. This is not possible
with r-sets.
Nevertheless, at least one has something mathematically precise on which to base
proofs.
5.3
Representation Schemes
Geometric modeling systems have taken many different approaches to representing
geometric objects. The following definitions ([ReqV82]) can be thought of as a start
towards being able to evaluate and judge these approaches in a rigorous way.
