Civil Engineering Reference
In-Depth Information
Figure 8.9
Surfaces combined to form a region.
non-manifold object with many internal partitions will be decomposed into regions,
each of which is bounded by individual surfaces so collected based on the smallest
dihedral angles between surfaces.
8.1.4.3 Validity check of the formation of regions
1. Topology correctness
The correctness in topology can be verified by the following three conditions:
a. Every region is closed.
b. Each surface is traversed two and only two times, one on each side.
c. For each region, check with the Euler-Poincare formula,
v
-
e
+
f
= 2(
s
-
h
)
where
v
= number of vertices,
e
= number of edges,
f
= number of faces,
s
= number of
surfaces (in this case, it is always equal to 1) and
h
= number of holes (genus), e.g. a torus
has one hole.
2. Geometrical Correctness
One of the geometrical checks that can be performed is to sum up the volumes of
individual regions
R
i
shown in Figure 8.10. Then
∑
, where N
R
is the
vol(
R
=
)
0
i
i
=
1,
N
R
number of regions so formed in the region construction process. If we define that the
vol(
R
1
) + vol(
R
2
) + vol(
R
3
) = 0
Internal surface
R
1
R
1
R
2
R
2
R
3
R
3
R
1
Figure 8.10
Sums of volumes of regions equal to zero.
