Civil Engineering Reference
In-Depth Information
H
6
Line segment L
H
4
H
2
Ω
4
Ω
6
Ω
1
Ω
2
Ω
3
Ω
5
H
1
H
3
H
5
Figure 2.44
Cell bounded by hyper-planes H
1
-H
6
.
The intersection between cell Ω and a line segment L can be obtained by considering indi-
vidual portions of the line segment cut off by the hyper-planes of the cell
L
∩=∩∩∩∩∩∩
=∩∩∩∩∩ ∩
L
L
(
)
1
2
3
4
5
6
(
)
(
L
)
(
L
)
(
L
∩∩∩∩∩
)
(
L
)
(
L
)
1
2
3
4
5
6
=∩∩∩∩∩
LL LLLL
1
2
3
4
5
6
L
i
= L ∩ Ω
i
= portion of L cut by Hi
i
If the original line segment L is mapped onto the interval [0,1], then intersection of line
segments L
1
, L
2
, …, L
6
can be easily done on this closed interval [0,1]. Whenever the result-
ing portion is reduced to an empty set during the process, there is no intersection between
the cell and the line segment L.
2.7.4 Determine the cells intersected by a triangular facet
To find the intersection between two solid objects, it is necessary to consider the intersec-
tion of their boundary surfaces, which are usually discretised into triangular facets. A 3D
background grid can be employed to speed up the searching of all potential triangular ele-
ments for intersection within the size range of a given triangular facet. The cells intersected
by triangular facet P
1
P
2
P
3
can be found as follows.
i. Calculate the limits of the triangular facet
a
min
= min(a
1
, a
2
, a
3
), b
min
= min(b
1
, b
2
, b
3
), c
min
= min(c
1
, c
2
, c
3
)
a
max
= max(a
1
, a
2
, a
3
), b
max
= max(b
1
, b
2
, b
3
), c
max
= max(c
1
, c
2
, c
3
)
where
P
1
= (a
1
, b
1
, c
1
), P
2
= (a
2
, b
2
, c
2
), P
3
= (a
3
, b
3
, c
3
).
As shown in Figure 2.45, cells potentially intersected by triangle P
1
P
2
P
3
are given by
(I
x1
→ I
x2
) × (I
y1
→ I
y2
) × (I
z1
→ I
z2
)
