1 Introduction
An optimal polygonization of a set of points in the plane is a classical problem in computational geometry and has been applied to many fields such as image processing marchand1999binary ; pakhira2011digital
pakhira2011digital ; pavlidis2013structural ; abdi2009effective , geographic information system galton2006region , etc. Considering a set of points in the plane, there are different numbers of simple polygons on . Enumerating and generating simple polygons on has been the focus of many studies zhu1996generating ; nourollah2017use ; garcia2000lower ; wettstein2014counting ; meijer1990upper .Finding polygons with special properties over all polygonizations is of particular interest to researchers. The minimum and maximum area polygonization are NPcomplete, as shown by Fekete fekete1993area ; fekete2000simple . The problems of computing the simple polygons with minimum and maximum perimeters is the wellknown NPcomplete problems called TSP and maxTSP, respectively. There are many ongoing studies on approximation algorithm for minimum and maximum area polygonization taranilla2011approaching ; peethambaran2016empirical , TSP bartal2016traveling ; moylett2017quantum and maxTSP dudycz20174 .
In some of this approaches the angles have been investigated in many problems over polygonization. The AngularMetric TSP aggarwal2000angular is the problem of finding a tour on minimizing the sum of the direction changes at each point. Fekete and Woeginger introduced AngleRestricted Tour problem in fekete1997angle . For a set of angles, AngleRestricted Tour is the problem of finding a simple or nonsimple polygon on where all angles of the polygon belong to . In asaeedi2017alpha concave hull refers to a simple polygon with minimum area covering a set of points such that all angels of are less than or equal to .
Reflexivity, the smallest number of reflex vertices among all polygonizations of a set of points, is considered as a convexity measurement for those points. Arkin et al. arkin2003reflexivity introduced the concept of reflexivity and presented lower and upper bounds for reflexivity of any set of n points. E. Ackerman et al. ackerman2009improved improved the upper bound and proposed an algorithm to compute polygon with at most this number of reflex vertices in the time complexity of . In lien2008approximate a convexity measurement has been proposed for polyhedra.
Rorabaugh citation0 investigated the minmax value of reflex angles in polygonizations as another convexity measurement for a set of points and derived an upper bound for their solution.
Here, we propose an upper bound for minmax value of the angles in polygonization and demonstrate that this bound is tight. Based on our knowledge to date, much less attention has been paid to this aspect so far rorabaugh2014bound . The rest of the paper is as follows: In the section 2, notations, definitions and some basic lemma are presented. In section 3, the upper bound is derived and in section 4, we conclude the paper highlighting its achievement.
2 Preliminaries
Let be a set of points in the plane and be the convex hull of . The vertices and edges of are denoted by and , respectively. Furthermore, let be the inner points of such that . Table 1 shows more notations that are used in the rest of the paper. A polygon crossing is specified by a closed chain of vertices such that .
Notation  Description 

A set of points in the plane  
cardinality of  
th point of ()  
convex hull of  
number of vertices of  
inner points of  
cardinality of  
vertices of  
edges of  
th vertex of ()  
th edge of ()  
a simple Polygon crossing  
an edge of with and as its end points (, )  
set of all simple polygons crossing  
, , ,  angles between 0 and 
Let be a line segment. The minor arc with measure equal to is denoted by , and the major arc with measure equal to is denoted by . Also, we denote the minor and major segments on by and , respectively (see Fig. 1).
Definition 1.
Let be a line segment. A sweep arc on is a minor arc where it expands to the major arc . Fig. 2 depicts the sweep arc on the line segment .
Lemma 1.
Let be a point inside the convex polygon , be the edges of and . Then, such that .
Lemma 2.
Let be a convex polygon, be the edges of and . The entire is covered by all major segments with measure equal to that correspond to the edges of , i.e. , (see Fig. 3).
Let be a set of points in the plane and suppose that the convex hull of has edges. Based on Lemma 2, is an upper bound for over all simple polygons crossing . It is noteworthy that this bound is tight. The tightness is achieved when the inner point is at the center of a regular ngons, as illustrated in Fig. 4.
3 MinMax Angle
In this section we present two upper bounds for and two algorithms to compute polygons satisfying the bounds. Let us first present some lemmas as follows.
Lemma 3.
Let be a line segment and be a set of points inside the , such that for an integer number . Assume that points are met by the sweep arc on and is a simple polygon such that all internal angles of are greater than or equal to . Let be th point met by the sweep arc. There exists an edge of such that is greater than or equal to .
Lemma 4.
Proof.
We prove Lemma 4 by constructing the polygon , using the following algorithm:
Algorithm 1 (Sweep Arc Algorithm)

Sweep the arc from to .

Let be the first point which is met by the sweep arc. Construct as the desired polygon.

Set .

Let be the constructed polygon inside the sweep arc and be the th point which is met by the sweep arc.

Assume that , , … , and are the edges of . If is visible from , set The angle subtended by at the point , otherwise set .

Let and be the edge that corresponds to .

Remove the edge from and add two edges and to construct the desired polygon.

Set . If , then go to 4, otherwise exit.
Based on Lemma 3, the angles in are greater than or equal to in step 4 of the algorithm. Therefore, when , the angles in are greater than or equal to . ∎
We refer to the constructed polygon by sweep arc algorithm, as a polygon corresponding to the line segment . In the following, based on Lemma 4, we present an algorithm to generate a polygon containing a given set of points such that all internal angles are less than .
Theorem 1.
There exists a polygon in which all internal angles of are less than or equal to .
Proof.
Here, by presenting algorithm 2 we construct the polygon.
Algorithm 2

Compute as the convex hull of and let be the set of inner points of .

For each edge of :

Compute the polygon corresponding to the edge using sweep arc algorithm to meet points of .

Remove vertices of from .


For all , the edges of minus all edges of except those that have no corresponding polygon, construct the desired polygon.
Based on Lemma 2, the entire is covered by where . Since the number of points inside the major segments are less than and also based on Lemma 4, all internal angles of the corresponding polygons are greater than or equal to . Hence, all internal angles of the polygon computed by algorithm 2 are less than . ∎
In step 2.a of algorithm 2, for each edge of , the measure of sweeping arc expands from 0 to and the sweeping arc contains the inner points as much as possible. In algorithm 3, presented below, the sweeping arcs that correspond to all edges of expand concurrently to contain all inner points. In this way, the upper bound is improved to such that is the depth of angular onion peeling on which is defined as follows.
Let us increase the measure of all sweeping arcs concurrently from 0 to the first hit (or , if a sweeping arc does not hit any point). All inner points that are hit by sweeping arcs form the layer 1 of points. The next layers are formed by deleting the points of the computed layer from inner points and keep increasing the measure of all sweeping arcs to the next hit. The process continues until all inner points are hit. The process of peeling away the layers, described above, is defined as ”angular onion peeling” and the number of layers is called ”depth of angular onion peeling” on these points.
Theorem 2.
There exists a polygon such that all internal angles of are less than where denotes the depth of angular onion peeling on .
Proof.
Here, by presenting algorithm 3, we construct such a polygon.
Algorithm 3

Compute as the convex hull of and let be the set of inner points.

While is not empty:

Increase the measure of all sweeping arcs to the next hit or .

Reconstruct the polygons corresponding to each edge of using the sweep arc algorithm.

Remove the visited points from .


All edges of the corresponding polygons computed in step 2 minus all edges of except those that have no corresponding polygon, construct the desired polygon.
Since the number of points inside the major segments are less than , all internal angles of corresponding polygons are greater than or equal to . Hence, all internal angles of the polygon computed by algorithm 3 are less than . ∎
Since the time complexity of sweep arc algorithm is , those of both algorithm 1 and 2 are . Based on Theorem 1, is an upper bound for . This bound is improved to in Theorem 2. When is a set of points in the plane and the convex hull of has edges, the depth of angular onion peeling on is equal to 1. Hence, the upper bound for is equal to which confirms the Remark 2.
Computing concave hull on a set of points is an NPcomplete problem asaeedi2017alpha . For all , concave hull crosses all points of . So, the polygon computed by algorithm 3 is an polygon asaeedi2017alpha which approximates concave hull of . The following corollary shows the relation between concave hull and the computed upper bound.
Corollary 1.
Let be a set of points in the plane, be the convex hull of , be the cardinality of edges of and be the depth of angular onion peeling on . For all , there always exists an concave hull on such that crosses all points of .
Coverage path planning is a fundamental problem in the field of robotics. There are many limitation factors in order to plan a path for a robot to cover (or visit) all points of a set of points, such as robot rotation angle. The following corollary presents the essential relation between path planning in robotics and our upper bounds on .
Corollary 2.
Let be a set of points in the plane, be the convex hull of , be the cardinality of edges of and be the depth of angular onion peeling on . If the robot rotation angle is greater than , there always exists a path for the robot to cover . As stated before, this path can be found in . Moreover, this rotation angle is tight.
4 Conclusion
The major problem investigated in this paper is that of finding a simple polygon with angular constraint on a given set of points in the plane. We derived the upper bounds for minmax value of angles over all simple polygons crossing the given set of points. We also presented new polynomial time algorithms to compute the polygons thereby satisfying the derived upper bounds. In addition to the theoretical results, this bound is an important achievement in the field of robotic.
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