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

It is to be noted that when both S and E lie on the same concave portion, then the SIP is
the portion of the curve C(t) between the two points [[3], Theorem 2.7, p220]. Also, straight
portions of SIP are tangential to the concave portions [18].
Definition 4 A BT is said to be consistent with respect to a path if it preserves the direction
of traversal of the path as stated in Lemma 2 .
It is also to be noted that, only one SIP is possible for a SCD, joining two distinct points,
lying on the boundary or interior [[21], Theorem 6.1, p147], which does not hold good for a
MCD.
Suppose that there is an SIP for a set curves constituting MCD. If an inner loop is added,
it should be observed that the entire SIP can change altogether. This suggests that an
incremental construction may not be possible and one has to consider all possible inner loops
at the same time. Moreover, local shortest path cannot be fine tuned to get the global path.
For example, a shortest path between two inner loops may not play a role at all in the global
shortest path between S and E (even-though those two curves may contribute to the final
path). Hence, all possible paths have to be detected to compute SIP.
Lemma 3 indicates the interaction that a path can have and the cases for which the path
can be eliminated, reducing the number of candidates and also computation of further BTs
for identifying SIP.
Lemma 3 (Path Elimination): Let there be a path. Let there be extended PCTs from S or
extended BTs consistent with the direction of traversal of the path. The path is eliminated
if it intersects any extended PCT from S or extended BT originating from the same concave
portion through which the path has passed.
Proof Let Fi represent the starting footpoint of BTi. Let η consist of the PCT, concave
portion between T and F1, and BT1 itself (refer to Figure 1(a)). Let the the bitangents BT2,
and BT3 arise out of the same concave portion as that of BT1. Let EBT2 and EBT3 represent
the intersection point of a path with their respective extended BTs.
Case 1: If η intersects any extended PCT from S at some point, say X, then the portion of
η till X can be replaced by the straight line portion SX which would make the path shorter
and hence η cannot be a potential SIP.
Case 2: Let the footpoint of a bitangent lies between T and F1 (such as F2 of BT2). Let
the path ζ = η + ζa and ζa intersects extended BT2 at EBT2. Then the cumulative portion
formed by concave portion of F2F1, BT1 and ζa can be replaced by straight line portion from
F2 to EBT2 to get a shorter path. Hence ζ can be removed.
Case 3a: Let the footpoint of a BT lies outside of T and F1 (such as F3 of BT3). Let the
path ζ = η + ζc. BT1 + ζc is longer than the path (F1F3 + F3 to EBT3) by the convex hull
property. Hence ζ can be removed.
Case 3b: Let the path ζ = η + ζb. The path ζ has self-intersection and hence it can be
eliminated.
Lemma 4 Let ζ be a path from S to E. Let G ∈ ζ be a point on a concave portion of one of
the input curves. Let SG and GE denote the portions of ζ before and after G respectively.
4