Civil Engineering Reference
In-Depth Information
7.5.5 Procedure for surface decomposition
7.5.5.1 Read in the surface S and carry out some
basic topological computation
Input
S
= {S
i
= (
v
1
,
v
2
,
v
3
)
i
, i = 1, N
S
}, {
p
i
= (x
i
, y
i
, z
i
), i = 1, N
p
}
where N
S
and N
p
are, respectively, the number of triangles and the number of points in
S
.
Find all the edges on the surface
S
and the two triangles connected to each edge,
E
= {(E, Δ
1
,
Δ
2
)
i
, i = 1, N
E
, Δ
1
, Δ
2
∈
S
}. Find the neighbours of each triangular facet S
i
,
T
= {(T
1
, T
2
, T
3
)
i
,
i = 1, N
S
; T
1
, T
2
, T
3
∈
S
}. T
i
is set to zero when there is no neighbour at a boundary edge.
The boundary of
S
is given by the collection of edges supported by only one triangular facet.
If no such edges exist, then S is a closed surface.
7.5.5.2 Determination of the cutting zone
The cutting zone is composed of a band of triangular facets intersected by the cutting plane.
Once cut plane P is determined, the set of triangular facets intersecting with the cut plane
P is given by I = {S
i
∈
S
, S
i
∩ P ≠ ∅}. A line segment AB is intersected by a cut plane P with
origin O normal to unit vector
u
if points A and B are on the opposite sides of the plane, as
shown in Figure 7.38a, i.e. (OA ⋅
u
)(OB ⋅
u
) ≤ 0. By adjacency relationship, in general, the
cutting zone is composed of strips and loops of intersected triangular facets, as shown in
Figure 7.38b and Section 4.5.4. Using the cutting zone as the natural separation line, the
surface will be automatically partitioned into a number of smaller pieces with the cutting
zone as boundaries between surface parts, as shown in Figure 7.39.
B
u
Cut plane P
O
(a)
Edge AB intersected
by cut plane P
A
Cutting zones
(b)
Lines of
intersection
Loop of triangles Chain of triangles
Figure 7.38
Determination of cutting zone: (a) edge AB intersected by cut plane P; (b) triangles intersected
form a loop or a chain.
Intersected triangles
Figure 7.39
Cut line is formed by edges.
