Civil Engineering Reference
In-Depth Information
ume (Aftosmis et al. 1998; Shostko et al. 1999). For the intersection of a pair of triangular
facets, four general cases can be identified as follows:
1. The intersection point P is inside triangle ABC, as shown in Figure 4.72a.
2. The intersection point P is on an edge of triangle ABC, as shown in Figure 4.72b.
3. The intersection point P is on a vertex of triangle ABC, as shown in Figure 4.72c.
4. The intersection is more than one point and is a planar region, as shown in Figure
4.72d.
Intersections are first classified into one of these four types, which will then be treated
accordingly. This classification enables a consistent treatment for the same intersection on
the two surfaces.
4.5.4 Tracing neighbours of intersecting triangles
As explained before, the intersection between surfaces is best represented by structural ele-
ments of chains and loops rather than individual unconnected line segments. Along each
intersection line segment, the intersecting triangles on each surface are neighbours to one
another. Making use of this neighbouring relationship, an intersection line can be con-
structed by tracing neighbouring triangles one after the other.
In the neighbour-tracing process, the type of intersection will indicate to which neighbour
the intersection line will extend. The following are the details of how neighbouring triangles
are traced for the four types of intersections: (1) inside the triangle, (2) on an edge, (3) at a
vertex and (4) planar zone of intersection.
1. If the intersection point is inside a triangle, there is no need to trace its neighbour. As
shown in Figure 4.73a for intersection type 1, point P is inside triangle S
1
. In this case,
there is no need to trace for the neighbour of S
1
. Instead, P is on the common edge of
T
1
and T
2
; hence, by neighbour tracing, the intersection line continues into T
2
.
2. As shown in Figure 4.73b, the intersection point P is on an edge of S
1
. The triangle
having this common edge with S
1
is S
2
, and the intersection line grows from S
1
to
neighbouring triangle S
2
.
3. If the intersection point is on a vertex of triangle S
1
, all the triangles connected to
this node have to be examined. Consider the intersection between triangles S
1
and T
1
,
as shown in Figure 4.73c. The intersection point P is on a vertex of S
1
; triangles S
2
,
S
3
, S
4
and S
5
having P as a common vertex will all be examined for intersection with
triangle T
1
.
S
2
T
2
S
1
I
S
2
T
1
P
S
3
S
1
Q
T
1
P
S
1
P
S
4
P
T
1
S
1
T
1
S
5
J
(a)
(b)
(c)
(d)
Figure 4.73
Construction of intersection line by tracing of neighbours: (a) type 1; (b) type 2; (c) type 3;
(d) type 4.
