The shortest path computation is important in industrial automation, especially for robot and autonomous vehicle navigation. However, most of the computations concentrate on computing the shortest path between two points within a polygon. The common approach for handling a bounded domain with free form boundary is to convert the domain into a polygon by boundary approximation so that the conventional computing algorithms can be used. Such an approximation affects the accuracy of the path. This article presents an approach to compute the shortest path between two given points in a free form boundary domain without any boundary approximation. This is addressed geometrically by imaginably placing a source at one of the points which radiates the shortest paths to various points of the domain. Some shortest paths are deflected by the geometry of the boundary so that they are no longer straight lines. Based on the deflections of the shortest paths, the bounded domain is partitioned into a set of subdomains. A tree is then constructed to show the relationships among these subdomains. The shortest path between two points is obtained from this tree.

## References

References
1.
Rohnert
,
H.
,
1986
, “
Shortest Paths in the Plane With Convex Polygonal Obstacles
,”
Inf. Process. Lett.
,
23
(
2
), pp.
71
76
.10.1016/0020-0190(86)90045-1
2.
Toussaint
,
G. T.
,
1986
, “
Shortest Path Solves Edge-to-Edge Visibility in a Polygon
,”
Pattern Recogn. Lett.
4
(
3
), pp.
165
170
.10.1016/0167-8655(86)90015-2
3.
Ghosh
,
S. K.
, and
Mount
,
D. M.
,
1987
, “
An Output Sensitive Algorithm for Computing Visibility Graphs
,”
Proceeding SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
, pp.
11
19
.
4.
Hershberger
,
J.
, and
Suri
,
S.
,
1999
, “
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
,”
SIAM J. Sci. Comput.
,
28
(
6
), pp.
2215
2256
.10.1137/S0097539795289604
5.
Kapoor
,
S.
,
Maheshwari
,
S. N.
, and
Mitchell
,
J. S. B.
,
1997
, “
An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane
,”
Discrete Comput. Geom.
,
18
(
4
), pp.
377
383
.10.1007/PL00009323
6.
Ling
,
H.
, and
Jacobs
,
D. W.
,
2007
, “
Shape Classification Using the Inner-Distance
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
29
(
2
), pp.
289
299
.
7.
Lee
,
D. T.
, and
Preparata
,
F. P.
,
1984
, “
Euclidean Shortest Paths in the Presence of Rectilinear Barriers
,”
Networks
,
14
(
3
), pp.
393
410
.10.1002/net.3230140304
8.
Melissaratos
,
E. A.
, and
Souvaine
,
D. L.
,
1992
, “
Shortest Paths Help Solve Geometric Optimization Problems in Planar Regions
,”
SIAM J. Sci. Comput.
,
21
(
4
), pp.
601
638
.10.1137/0221038
9.
Bourgin
,
R. D.
, and
Howe
S. E.
,
1993
, “
Shortest Curves in Planar Regions With Curved Boundary
,”
Theor. Comput. Sci.
,
112
(
2
), pp.
215
53
.10.1016/0304-3975(93)90019-P
10.
Fabel
,
P.
,
1999
, “
Shortest Arcs in Closed Planar Disks Vary Continuously With the Boundary
,”
Topol. Appl.
,
95
(
9
), pp.
75
83
.10.1016/S0166-8641(97)00275-7
11.
Ram
,
S. B.
, and
Ramanathan
,
M.
,
2012
, “
The Shortest Path in a Simply-Connected Domain Having a Curved Boundary
,”
Comput.-Aided Des.
,
43
(
8
), pp.
923
933
.
12.
Li
,
F.
, and
Klette
,
R.
2006
, “
Finding the Shortest Path Between Two Points in a Simple Polygon by Applying a Rubberband Algorithm
,” The Pacific-Rim Symposium on Image and Video technology, pp.
280
291
.10.1007/11949534
15.
Boissonnat
,
J. D.
, and
Teillaud
,
M.
,
2006
,
Effective Computational Geometry for Curves and Surfaces
,
Springer
.