Civil Engineering Reference
In-Depth Information
Construction
front
Surface
boundary
Seed
triangle
Open surface
Closed surface
Construction front reduced to zero
Figure 8.5
Surface construction.
Let n(L) be the number of triangles connected to edge L; then L is a boundary edge if
n(L) ≠ 2. No search is needed since the elements connected to each edge are known and well
prepared beforehand. Efficiency of the scheme is of order N
B
. The procedure guarantees that
each surface generated is orientable (non-orientable surfaces such as Mobius strip can be
detected when there is an inconsistency in the orientation between neighbouring triangles).
All the elements belonging to the same surface will be given a unique surface label and
oriented with respect to the first (seed) element of the surface. Elements used in the surface
construction will be deleted from set
B
, and the surface-construction process terminates if
there are no more triangles in
B
. Each triangle in
B
belongs to one and only one surface, i.e.
B
is effectively partitioned into surface parts by the surface-construction procedure. If N
S
is
the number of surfaces so created, then N
S
≤ N
B
.
8.1.3.5 Flagging unused surface parts
All hanging surface parts with free edges, n(L) = 1, are flagged, and they will not be used
in the next step of region identification. In 3D MG, these surface parts may represent addi-
tional constraints in triangulation.
8.1.4 Region identification
8.1.4.1 Boundary edges
Find the boundary edges of each surface. Boundary edges exist for open surfaces, whereas for
closed surfaces such as sphere and torus, there is no boundary edge, as shown in Figure 8.6.
8.1.4.2 Formation of regions
1. Start with any open surface; take note of the two orientations (sides) of the surface, as
shown in Figure 8.7a.
2. Pick up any free edge L on the unclosed surface; join it with the other surface (such a
surface exists as n(L) > 2) at the free edges for which the dihedral angle θ between the
surfaces is the smallest, as shown in Figure 8.7b. The 2D analogy is shown in Figure
8.8; regardless of the complexity of the internal region, the boundary loop can always
