Civil Engineering Reference
In-Depth Information
∂
. To determine the intersection between Ω and , it is much simpler to concentrate on
their boundary, and it turns out that it is also sufficient to determine the regions or vol
u
mes
of intersection by considerin
g
solely the intersection of their boundaries, i.e. ∂Ω and
∂
.
When ∂Ω interacts with
∂
, the result is a finite number of closed loops of line segments,
as shown in Figure 8.105. The loops are non-intersecting and distinct as boundary surfaces
are smooth and closed, such that valid intersections on the boundary surface can always be
represented by loops of intersection segments. Even in the worst case where loops sharing
a common point as in the case of the intersection of cylindrical surfaces of equal diameter,
the situation could be easily resolved using the non-intersecting principle by accepting an
intersection loop composed of two half circles as shown in Figure 8.106.
The intersection loops due to the intersection of the two boundary surfaces can be
conveniently determined in a robust manner by the method of neighbour tracing making
the best use of the property of continuity of intersection lines, as discussed in Section
4.5.4. Along each intersection line segment, the intersecting triangles on each surface are
neighbours to each other to form a closed chain, as shown in Figure 8.107. From the chain
of triangles that contain all the intersection segments, by means of the neighbouring rela-
tionship, an intersection loop can always be constructed by tracing neighbouring triangles
one after the other. The reliability of the method is greatly enhanced as intersection can be
locally controlled and guided by the adjacency relationship, and a clear direction could be
specified as how an intersection loop should progress from one triangular facet onto the
next. Most importantly, the neighbour-tracing method ensures that a closed loop of inter-
section could always be defined on a closed surface starting from any intersection segment
between a pair of intersecting triangles. As the neighbour-tracing process is carried out
Figure 8.105
Intersection between a sphere and a cylinder.
Intersection
loop
(a)
(b)
Figure 8.106
Intersection of cylindrical surfaces: (a) equal diameter; (b) unequal diameter.
