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

C1
I1
I2
CSM
(a) An example of interior
region elimination
C2
W
4.1.3 Region-based elimination
E E
(b) PCTs from S after elimination
Figure 4: Valid PCTs from S, E and interior region
S
(c) Termination list from E
Once the regions are identified, all PCTs and curves that are contained in the regions are
eliminated (Lemma 4). In order to get the PCTs within the regions, the values counterclockwise
angle with respect to the SE line or parameter values of IOC are used. Since the
PCTs are from the same starting point, the angle and the parameter of IOC correlate. Firstly
a range of parametric values is identified using the PCTs that contribute to the identification
of regions. Then tangents falling within or outside the range are eliminated accordingly. For
curve elimination within a region, typical intersection check followed by ray shooting is done.
Figure 4(b) and 4(c) show the remaining PCTs from S and E after employing region-based
elimination. Inner loops that are eliminated using exterior regions R1 and R2 are shown as
gray in Figure 3(b). Algorithm 1 describes the pseudocode for obtaining PCTs and processing
them.
4.2 Processing BTs
BTs are computed from each PCT in the starting list (which contains thus far processed
PCTs, Figure 4(b)). A similar region elimination strategy is followed for BTs with some
modifications to the region identification. Consider a path ζ, shown in Figure 5(a) from S
to a point T on a curve CT G, which is an incomplete potential SIP (i.e., the path has not
yet reached the end point E). At and near T , CT G is concave. Let φ (clockwise or counterclockwise)
be the direction induced by ζ on CT G. Let NIF be the closest inflection point to
T on CT G in the direction φ. Now T and NIF identify a concave portion of CT G.
To identify the next set of potential paths for ζ, the set of bitangents from concave portion
T NIF is computed. The bitangents that are not completely contained in MCD and those that
are not consistent with direction φ are removed. Let the starting foot point of each bitangent
be denoted by FST . The starting curve, CST , for all bitangents is CT G. Let the end foot point
and corresponding end curve be denoted by FEN and CEN. Each bitangent is extended from
8