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

E
S
(a) Sample set of curves (b) PCTs that are
completely contained in
MCD
Figure 2: Sample set of curves and PCTs from S
(the footpoint denoted as T ). Let CF E denote the curve at which the extended PCT intersects
first, with the intersection point denoted as IF E. Also, let IOC be the first intersection of the
PCT with the outer curve COC. It is to be noted that the extended PCT can intersect the
outer curve first and in that case, CF E = COC, and IF E = IOC.
4.1 Processing PCTs
Initially, PCTs from S (using constraint equation in [18]) are used as the starting tangents.
All the PCTs that are not completely contained in the MCD are then removed (Figure 2(b)
shows only PCTs that are completely contained in the MCD). Considering the Figure 3(a),
for the PCT ST1, the curve C1 is CT G which has the footpoint T1. The extended PCT first
intersects at P1. Hence IF E = P1. Since P1 is at the outer curve COC, IF E = IOC = P1
and CF E = COC. Note that T1P1 is completely contained in the MCD. For the extended
PCT whose first intersection IF E = I3 with curve CF E = C3, CT G = COC. However, for the
extended PCT whose first intersection IF E = I4 with curve CF E = C3, CT G = C5. All the
extended portions of the PCTs are identified by dashed lines.
4.1.1 Exterior region identification
Among the PCTs that are completely contained in MCD, a PCT with its footpoint T or IF E
lying on COC, the closest in clockwise direction along the curve COC to the end point, E, is
selected. Though the search is done in clockwise direction, the closest point (for example,
P3 in Figure 3(a)) is in the counter-clockwise portion between S and E, when travelled from
S. Hence the PCT is named as CCRP CT (counter-clockwise region PCT). For a PCTs thus
selected, the following regions are described based on the conditions delineated as follows.
1. If CT G and CF E are both COC respectively, then two regions are identified. One region
consisting ST and portion of outer curve from T to S (that does not contain E) as
6
S