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

non-intersecting polygonal obstacles (e.g. [14, 19]). A prominent approach to compute the
shortest path for polygonal obstacles is via the visibility graph construction [2] as shortest
path is a subset of the visibility graph.
Shortest interior path (SIP) in a curved boundary can be thought of as a combination
of Euclidean and geodesic, where the SIP will consist of straight lines and arcs of curved
boundaries [18]. Approximating curved boundaries with polygons generates inaccuracies in
SIP and is not satisfactory in general [7]. Another approach to compute SIP in a curved
boundary is to sample points on the curved boundary [12] and then employ a typical shortest
path algorithm (such as Dijkstra [1]). The accuracy will depend on the sampling technique
used and is an approximate computation of the SIP. Algorithm for piecewise-linear boundaries
such [11] has also been extended to curved boundary using “funnels” in [13] assuming the
input (curved boundary) is already triangulated. Numerical approximation of shortest path
in curved obstacles using fast-sweeping of Eikonal equations [22] and grid-based approach has
been presented in [17].
Very few algorithms have handled the computation of SIP of curved boundary without
approximating with polygons or sample points, albeit theoretically. Computing visibility
graph based on greedy approach pseudo-triangulation for a set of non-intersecting convex
curved objects has been presented [16]. A similar approach for convex curves that intersect
transversely has been given in [5].
For arbitrarily shape curved boundary (not just convex), Bourgin and Howe [3] presented
an algorithm using string formulation on the intersection of the line connecting the start and
end points with the curved boundary. Based on the maximal distance of the curve from the
line, different cases are delineated to find the points on the curve that belong to the SIP.
Fabel [10] has presented an algorithm for computing the shortest arcs in a closed planar disk.
Both [3] and [10] are theoretical descriptions and no implementation results are provided.
An algorithm with implementation results for simply-connected arbitrarily shaped curved
boundary has been presented in [18] using tangent computation and region check. The major
idea in computing the SIP of curved boundaries [3, 10, 18] is to avoid computing visibility
graph through identifying potential straight lines/arcs of the curves using strategies such as
maximal distance [3] , region check [18] etc.
To the best of our knowledge, no known algorithm with implementation results is available
for arbitrarily shape curved boundary of a multiply-connected domain (MCD). In this paper,
the aim is to find the shortest interior path (SIP) of a MCD having curved boundaries,
extending the algorithm presented in [18], using the basic idea of tangent computation. All
the possible paths are directly identified during the computation effectively eliminating the
use of algorithms such as Dijkstra. SIP is then identified from the set of paths. The following
are the key differences from [18].
• In case of simply-connected domain (SCD) [18], the focus was on computing the SIP
itself, since SIP was unique. As there will be number of paths that could contribute
towards SIP in the MCD, in this paper, the focus is on eliminating a path that will not
play a role in the set of potential paths. Lemma 3 in the paper indicates the interaction
that a path can have and the cases for which the path will cease to exist.
• In [18], line connecting S and E was used to divide the boundary curve into two portions.
As such a division is not possible in multiply-connected domain (MCD), Lemma 4 in the
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