Civil Engineering Reference
In-Depth Information
Distance = 0
Front
d
i
at
p
i
r
i
p
Compute distance d at
p
Figure 7.42
Distance map.
determine the geodesic distance point by point, it is much easier to set up a distance map for
all the points by means of a recursive formula. The distance at a point
p
can be computed
from its neighbouring points, as shown in Figure 7.42, such that
d
=
min(
d
+
r
)
(7. 3)
i
i
ineighbour
∈
where d
i
is the distance of a neighbouring point
p
i
, and r
i
is the distance between
p
and
p
i
.
As the adjacency relationship of the triangular facets has already been determined, the
distance of all the points can be computed by means of a frontal process starting from the
cut line, where points are set a datum of zero value. Take a point
p
next to the front where
the distance from the datum line is known. Form the patch of triangles around
p
, as shown
in Figure 7.42, and the distance at
p
is calculated using formula 7.3 on all its neighbouring
frontal points with known distance from the datum. Point
p
is absorbed to the front as its
distance from the datum is now known. The front of points of known distance continues
to propagate until all the points are processed. This method does not produce the absolute
distance from the cut line but the pseudo-distance along the edges of the discretised surface.
However, the distance map so computed is accurate enough to differentiate the relative dis-
tances of two points from the cut line.
7.5.5.6 Marching on the surface
The original cut line can march forward (shift in one direction) by absorbing one or more trian-
gular facets (elements) nearest to the cut line. In the more general situation where the marching
process is based on some user-defined geometrical/topological criterion α, the smaller surface
expands by taking triangles from the large surface at the boundary between the two surfaces.
The balance criterion α of the surfaces has to be evaluated from time to time when more tri-
angles are taken up by one of the surfaces in the marching process. The final position of the
cut line is determined when the two surface parts are more or less balanced measured in α
quantity, as shown in Figure 7.43. In the marching process, the boundary will be smoother
Proposed cut line
Original cut line
Figure 7.43
Marching on surface.
