Civil Engineering Reference
In-Depth Information
n
n
Figure 2.16
Normal at a corner of a cube.
2.4.6 Normal at a node
The normal to a plane is well defined by the cross product of any two non-parallel edges,
and this definition has to be generalised to the normal at a point on a discretised surface.
Based on the notations of Section 2.4.5, as shown in Figure 2.15, the angle P
k
PP
k+1
of the k
th
triangular facet is given by
α
k
= cos
−1
(
u
k
⋅
u
k+1
)
The normal at node P is computed by the weighted average of the surface normals:
∑
∑
α
n
N
N
kk
N
=
and unit normal,
n
=
α
k
By this definition, the normal at a point depends only on the geometry of the surface
but not on how it is discretised into elements. For instance, the surface normal at a corner
of a cube is the same for the two surfaces meshed in two different patterns, as shown in
Figure 2.16.
2.4.7 Intersection between a line segment and a triangular facet
The following are the procedures for the determination of the intersection between line seg-
ment Q
1
Q
2
and triangle P
1
P
2
P
3
in space.
i. Compute the signed distance h
1
and h
2
of points Q
1
and Q
2
to the plane P
1
P
2
P
3
. If
h
1
h
2
< 0, calculate ξ = h
1
/(h
1
− h
2
). Intersection point Q on the plane is given by Q =
Q
1
+ ξ(Q
2
− Q
1
).
ii. Calculate the barycentre co-ordinates of Q, (L
1
, L
2
, L
3
). If L
1
≥ 0, L
2
≥ 0, and L
3
≥ 0,
Q is inside triangle P
1
P
2
P
3
and Q
1
Q
2
intersects P
1
P
2
P
3
, as shown in Figure 2.17.
L
3
Q
1
h
1
P
3
P
1
Q
L
1
h
2
P
2
Q
2
L
2
Figure 2.17
Intersection between a line segment and a triangular facet.
