scholarly journals Convexly Independent Subsets of the Minkowski Sum of Planar Point Sets

10.37236/883 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Friedrich Eisenbrand ◽  
János Pach ◽  
Thomas Rothvoß ◽  
Nir B. Sopher
Keyword(s):  
Minkowski Sum ◽  
Finite Sets ◽  
Point Sets ◽  
Planar Point

Let $P$ and $Q$ be finite sets of points in the plane. In this note we consider the largest cardinality of a subset of the Minkowski sum $S\subseteq P \oplus Q$ which consist of convexly independent points. We show that, if $|P| = m$ and $|Q| = n$ then $|S| = O(m^{2/3} n^{2/3} + m + n)$.

scholarly journals A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

10.37236/484 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ondřej Bílka ◽  
Kevin Buchin ◽  
Radoslav Fulek ◽  
Masashi Kiyomi ◽  
Yoshio Okamoto ◽  
...  
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Minkowski Sum ◽  
Independent Subset ◽  
Point Sets ◽  
Planar Point ◽  
Minkowski Sums

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function $M(m, n)$, which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets $P$ and $Q$ with $|P| = m$ and $|Q| = n$. They proved that $M(m,n)=O(m^{2/3}n^{2/3}+m+n)$, and asked whether a superlinear lower bound exists for $M(n,n)$. In this note, we show that their upper bound is the best possible apart from constant factors.

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 .

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 Point Pattern Matching Algorithm for Planar Point Sets under Euclidean Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaoyun Wang ◽  
Xianquan Zhang
Keyword(s):  
Pattern Matching ◽  
Complete Graph ◽  
Point Pattern ◽  
Point Sets ◽  
Matching Problem ◽  
Matching Algorithm ◽  
Planar Point ◽  
Point Pattern Matching ◽  
Point Set ◽  
Pattern Matching Algorithm

Point pattern matching is an important topic of computer vision and pattern recognition. In this paper, we propose a point pattern matching algorithm for two planar point sets under Euclidean transform. We view a point set as a complete graph, establish the relation between the point set and the complete graph, and solve the point pattern matching problem by finding congruent complete graphs. Experiments are conducted to show the effectiveness and robustness of the proposed algorithm.

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