Graphics Reference
In-Depth Information
(Exercise 5.2.2). In other words, r
X
is an r-set for any subset
X
of
R
n
. R-sets seem to
capture the notion of being a solid. Anything called a
solid
should be an r-set, but we
shall refrain from giving a formal definition of the word “solid.” In many situations,
one would probably want that to mean a compact (closed and bounded) n-manifold.
R-sets are more general than manifolds, however. The union of two tetrahedra which
meet in a vertex is an r-set but not a 3-manifold because the vertex where they meet
does not have a Euclidean neighborhood.
Because halfplanes are r-sets we get all our linear polyhedral “solids” from those
via the Boolean set operators such as union, intersection, and difference. We can think
of halfspaces as primitive building blocks for r-sets if we allow “curved halfspaces” by
extending the notion as follows:
A
halfspace
in
R
n
is any set of the form
Definition.
()
=
{
()
≥
}
( )
=
{
()
£
}
Hf
pp
f
0
or Hf
pp
f
0 ,
+
-
where f :
R
n
Æ
R
. If
H
is a halfspace, then we shall call r
H
a
generic halfspace
. A finite
combination of generic halfspaces using the standard operations of union, intersec-
tions, difference, and complement is called a
semialgebraic
or
semianalytic set
if the
functions f are all polynomials or analytic functions, respectively.
For example, the infinite (solid) cylinder of radius R about the z-axis, that is,
{
}
(
)
2
2
2
xyz x
,,
+- £
y
R
0
,
is a generic halfspace, in fact, a semialgebraic set. See Figure 5.3. Semialgebraic sets
are an adequate set of building blocks for most geometric modeling and are also “com-
putable” (see Section 5.11).
Next, we need to address a problem with the standard Boolean set operators,
namely, they are not closed when restricted to r-sets. For example, the intersection of
the two r-sets
X
= [0,1] ¥ [0,1] and
Y
= [1,2] ¥ [0,1] is not an r-set. See Figure 5.4.
From the point of view of solids, we would like to consider
X
and
Y
as being disjoint.
One sometimes calls
X
and
Y
quasi-disjoint
, which means that their intersection is a
lower-dimensional set. If we want closure under set operations, we need to revise their
definitions.
Figure 5.3.
A generic halfspace.
