Graphics Reference
In-Depth Information
Figure 5.9.
Primitive instancing scheme.
A primitive instancing scheme is only really useful when the number of primitives
is small. The scheme is similar to a language in which words cannot be combined to
make a sentence. Note that a good structured modeling system can achieve all the
advantages of a primitive instancing scheme by providing a macro capability.
5.3.2
Boundary Representations
Since one only sees the surface of solids, it is natural to represent them via their
boundary. Mathematically this is justified because in the special case of closed and
bounded solids in
R
n
the boundary of a solid uniquely defines that solid. (In general,
though, the boundary of a manifold does
not
define the manifold.)
Definition.
The
boundary representation
or
b-rep
of r-sets in
R
n
is the representation
that associates to each r-set
X
its boundary b
X
. More generally, a b-rep for r-sets is a
representation that associates to an r-set
X
a representation of b
X
.
B-reps are probably the most common representations used in computer graph-
ics. If
X
is a solid in
R
3
, then
ยป
"face" of
b
X
=
F
FX
.
The boundary of each face is itself a union of edges and the boundary of an edge is
a set of two points. One can therefore think of a boundary representation scheme for
solids in terms of a relation between objects and certain graphs. See Figure 5.10 for
the most basic version of such a graph. For linear polyhedra it is actually a function.
For curved objects it is not a function because it depends on how the boundary of the
object is divided into cells (curved patches).
There are some difficult questions having to do with the validity of boundary rep-
resentations. In particular,
When does a graph of the type shown in Figure 5.10 come from a real polyhedron?
