Let S be a set of  points in the plane, CH be the convex hull of S, \wp(S) be the set of all simple polygons crossing S, \gamma_P be the maximum angle of polygon P \in \wp(S) and \theta =min_{P\in\wp(S)} \gamma_P. In this paper, we prove that \theta\leq 2\pi-2\pi{r.m} such that m and r are the number of edges and internal points of CH , respectively. We also introduce an innovative polynomial time algorithm to construct a polygon with the said upper bound on its angles. Constructing a simple polygon with angular constraint on a given set of points in the plane is highly applicable to the fields of robotics, path planning , image processing, GIS, etc . Moreover, we improve our upper bound on \theta and prove that this is tight .
Let S be a set of n points in the plane, \wp(S) be the set of all simple polygons crossing S, \gamma_P be the maximum angle of polygon P \in \wp(S) and \theta =min_{P\in\wp(S)} \gamma_P. In this paper, we prove that \theta\leq 2\pi-2\pi{r.m} where m and r are the number of edges and inner points of the convex hull of S , respectively. We also propose an algorithm to construct a polygon with the said upper bound on its angles. Constructing a simple polygon with angular constraint on a given set of points in the plane can be used for path planning in robotics . Moreover, we improve our upper bound on \theta and prove that this is tight for r=1.
A
n
D
CH be the convex hull of S,
R
such that
where
R
internal points of CH
inner points of the convex hull of S
R
introduce an innovative polynomial time
propose an
R
is highly applicable to the fields of robotics, path planning , image processing, GIS, etc
can be used for path planning in robotics
R
.
for r=1.