Civil Engineering Reference
In-Depth Information
E
S
i
S
i
ℓ
C
B
Ω
θ
φ
A
D
S
j
S
j
Figure 8.114
Determination of cutting surfaces.
Whether a triangular facet connected to an edge is inside a volume or not can be deter-
mined by considering the volumes of the tetrahedra so formed, as shown in Figure 8.114,
where arrows are pointing towards the interior of the object. Let V
1
and V
2
be the volumes
of tetrahedra ABCE and BADE, i.e. V
1
= V(A, B, C, E) and V
2
= V(B, A, D, E). If both V
1
and V
2
are positive, then face ABE is between faces ABC and BAD, or E is inside the volume
bounded by faces ABC and BAD. On the contrary, if V
1
and V
2
are both negative, then point
E is outside of the volume bounded by faces ABC and BAD. However, if either V
1
or V
2
is
positive and the other negative, then the volume V = V(A, B, C, D) has to be checked. If V is
positive, E is outside, and on the other hand, if V is negative, E is inside the bounded volum
e
.
As each loop connects to exactly two cutting surfaces, one from ∂Ω and the other from
∂
,
between S
i
an
d
S
j
⊂ ∂Ω connected to loop
ℓ
,
suppose S
k
is the cutting surface into mesh ,
and similarly,
S
k
is the cutting surface in
∂
connected to
ℓ
cutting into v
ol
ume Ω. Check if
there are any more intersection loops connected to surface patches S
k
and
S
k
, and include all
these loops and surface patches to this region until no more surface patches and intersection
loops could be collected for this region. This loop and surface patch collection process is
well defined and will be completed rapidly as each intersection loop and each cutting surface
belong to one and only one volume of intersection. The collection process is repeated when
there are intersection loops remaining untreated without being assigned to any region of
V
1
S
4
S
4
S
2
S
1
S
5
Ω
S
1
S
2
S
3
Ω
S
3
Figure 8.115
Intersection between hollow cylinder and ellipsoid.
