Empty Convex Hexagons in Planar Point Sets

2008 ◽  
pp. 1-34 ◽  
Author(s):  
Tobias Gerken
Keyword(s):  
Point Sets ◽  
Planar Point ◽  
Empty Convex

Planar point sets with a small number of empty convex polygons

2004 ◽  
Vol 41 (2) ◽  
pp. 243-269 ◽  
Author(s):  
Imre Bárány ◽  
Pável Valtr
Keyword(s):  
Convex Hull ◽  
General Position ◽  
Point Sets ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Finite Set ◽  
Empty Triangles

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

scholarly journals Disjoint empty convex pentagons in planar point sets

2012 ◽  
Vol 66 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Bhaswar B. Bhattacharya ◽  
Sandip Das
Keyword(s):  
Point Sets ◽  
Planar Point ◽  
Empty Convex

scholarly journals EMPTY CONVEX 5-GONS IN PLANAR POINT SETS

2009 ◽  
Vol 31 (3) ◽  
pp. 315-321
Author(s):  
Seong-Yoon Ann ◽  
En-Sil Kang
Keyword(s):  
Point Sets ◽  
Planar Point ◽  
Empty Convex

scholarly journals Disjoint empty convex polygons in planar point sets

2001 ◽  
Vol 56 (2) ◽  
pp. 62-70
Author(s):  
Attila Guly�s ◽  
L�zl� Szab�
Keyword(s):  
Point Sets ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex

scholarly journals Empty Convex Hexagons in Planar Point Sets

2007 ◽  
Vol 39 (1-3) ◽  
pp. 239-272 ◽  
Author(s):  
Tobias Gerken
Keyword(s):  
Point Sets ◽  
Planar Point ◽  
Empty Convex

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

scholarly journals Planar point sets determine many pairwise crossing segments

2021 ◽  
Vol 386 ◽  
pp. 107779
Author(s):  
János Pach ◽  
Natan Rubin ◽  
Gábor Tardos
Keyword(s):  
Point Sets ◽  
Planar Point

scholarly journals MAINTAINING VISIBILITY INFORMATION OF PLANAR POINT SETS WITH A MOVING VIEWPOINT

2007 ◽  
Vol 17 (04) ◽  
pp. 297-304 ◽  
Author(s):  
OLIVIER DEVILLERS ◽  
VIDA DUJMOVIĆ ◽  
HAZEL EVERETT ◽  
SAMUEL HORNUS ◽  
SUE WHITESIDES ◽  
...  
Keyword(s):  
Linear Space ◽  
Point Sets ◽  
Worst Case ◽  
Planar Point ◽  
Set Of Points

Given a set of n points in the plane, we consider the problem of computing the circular ordering of the points about a viewpoint q and efficiently maintaining this ordering information as q moves. In linear space, and after O(n log n) preprocessing time, our solution maintains the view at a cost of O( log n) amortized time (resp.O( log 2 n) worst case time) for each change. Our algorithm can also be used to maintain the set of points sorted according to their distance to q .

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