Shortest path in a multiply-connected domain having curved ...

(a) All BTs in the given
shape (568 BTs)
(b) Completely contained
BTs (183 BTs)
Figure 10: Number of bitangents
though it cannot be quantified exactly how many tangents will be removed as it depends on
the shape of the curves. The number of BTs that will be used will be a substantial reduction
of the total number of BTs, when we compute on the fly over any algorithm that might use
all the completely contained tangents. Consider the input curves in Figure 7(a). The curves
have 568 BTs (Figure10(a)) and 183 completely contained BTs (Figure10(b)). By including
the PCTs, approximately there are 200 completely contained tangents. These are the input
for graph based Dijkstra’s algorithm. However, our algorithm has been found to use far
lesser number of tangents as shown in the Table 1. The table shows the number of tangents
selected/number contained tangents at each stage along the recursion. Blank entry in the
table means all paths have terminated.
Table 1: Number of tangents selected/Number contained tangents at each stage
Figure PCTs from E PCTs from S BTs BTs BTs BTs BTs BTs BTs
7(a) 4/11 5/12 2/8 1/10 1/6 1/12
7(c) 3/3 4/10 1/12 2/4 1/10 1/16 2/3 2/12 1/6
7(e) 3/14 6/10 3/11 2/12 1/15 2/16 1/12 1/12
It should be noted that application of Lemma 4 leads to inner loops and regions get
rejected at each stage, BTs are computed and used only in a closed band around the shortest
path as could be seen from figures 7(a) and 7(e). Thus the number of computations are also
greatly reduced.
5.2.2 Using discretization of the curve to compute SIP
Discretizing the curve into set of points or polylines and then employing Dijkstra’s algorithm
could be another approach to compute SIP. However, to compute a path, it is imperative that
the footpoints of the tangent (PCTs/BTs) were part of sample points or vertices of polylines.
However, there is no effective sampling strategy that can precisely capture all footpoints of
tangents and bitangents and hence more inaccuracies are introduced. As one has to anyway
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