Graphics Reference
In-Depth Information
Figure 5.10.
A boundary representation
graph.
There is no easy answer to this question. Here are some conditions in case the b-rep
for a solid is supposed to be induced from a simplicial decomposition, that is, the
solid is the underlying space of a simplicial complex:
(1) Each face must have three edges.
(2) Each edge must have two vertices.
(3) Each edge must belong to an even number of faces.
(4) Each vertex of a face must belong to precisely two edges of the face.
(5) All points (x,y,z) must be distinct.
(6) Edges must be disjoint or intersect in a vertex.
(7) Faces must be disjoint or intersect in edges.
Conditions (1)-(4) deal with the combinatorial topology of simplicial complexes and
are easy to check. Conditions (5)-(7) are point set topology questions that are expen-
sive to test.
Some common data structures that are used to implement the boundaries of linear
polyhedra are described in Section 5.8.1.
5.3.3
The CSG Representation
In constructive solid geometry (CSG) one represents objects in terms of a sequence
of Boolean set operations and rigid motion operators starting with a given collection
of primitive objects. One can express this representation pictorially as a binary tree.
For example, in Figure 5.11 the binary tree on the left is used to represent the union
of three blocks, one of which has been translated by a vector v . Although the idea is
simple enough, we must get a little technical in order to give a precise definition.
Let P be a set of r-sets in R n . The elements of P will be called primitive objects.
Let O be a set of regularized binary set operators such as »*, «*, -*,.... Let M be a
set of pairs (m,x) where m is a label for a rigid motion of R n such as a translation,
rotation, ..., and x is data that defines a specific such motion. For example, if n = 2,
then (rotation,( p ,p/2)) is a possible pair in M and represents the rotation of R 2 about
p through an angle p/2. If (m,x) ΠM, then let m(x) denote the rigid motion defined
by the pair (m,x).
Search WWH ::




Custom Search