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 A Bound on a Convexity Measure for Point Sets

2019 ◽  
Vol 29 (04) ◽  
pp. 301-306
Author(s):  
Danny Rorabaugh
Keyword(s):  
Upper Bound ◽  
Interior Angle ◽  
Point Sets ◽  
Planar Point ◽  
Angle Measure ◽  
Convex Position ◽  
Point Set ◽  
Set Of Points ◽  
Convexity Measure

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

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.

scholarly journals Counting Triangulations of Planar Point Sets

10.37236/557 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Micha Sharir ◽  
Adam Sheffer
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Line Graphs ◽  
Point Sets ◽  
Planar Point ◽  
Straight Line ◽  
Point Set ◽  
Spanning Cycles

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of $43^n$ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, it can be used to derive new upper bounds for the number of planar graphs ($207.84^n$), spanning cycles ($O(68.67^n)$), spanning trees ($O(146.69^n)$), and cycle-free graphs ($O(164.17^n)$).

scholarly journals Dense Point Sets with Many Halving Lines

2019 ◽  
Vol 64 (3) ◽  
pp. 965-984
Author(s):  
István Kovács ◽  
Géza Tóth
Keyword(s):  
Discrete Comput Geom ◽  
Lower Bounds ◽  
Higher Dimensions ◽  
General Point ◽  
Point Sets ◽  
Planar Point ◽  
Dense Point ◽  
Point Set ◽  
Dense Point Set ◽  
Dense Set

Abstract A planar point set of n points is called $$\gamma $$ γ -dense if the ratio of the largest and smallest distances among the points is at most $$\gamma \sqrt{n}$$ γ n . We construct a dense set of n points in the plane with $$ne^{\Omega ({\sqrt{\log n}})}$$ n e Ω ( log n ) halving lines. This improves the bound $$\Omega (n\log n)$$ Ω ( n log n ) of Edelsbrunner et al. (Discrete Comput Geom 17(3):243–255, 1997). Our construction can be generalized to higher dimensions, for any d we construct a dense point set of n points in $$\mathbb {R}^d$$ R d with $$n^{d-1}e^{\Omega ({\sqrt{\log n}})}$$ n d - 1 e Ω ( log n ) halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.

ON FLIPS IN POLYHEDRAL SURFACES

2002 ◽  
Vol 13 (02) ◽  
pp. 303-311 ◽  
Author(s):  
OSWIN AICHHOLZER ◽  
LYUBA S. ALBOUL ◽  
FERRAN HURTADO
Keyword(s):  
Finite Point ◽  
Polyhedral Surfaces ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set

Let V be a finite point set in 3-space, and let [Formula: see text] be the set of triangulated polyhedral surfaces homeomorphic to a sphere and with vertex set V. Let abc and cbd be two adjacent triangles belonging to a surface [Formula: see text]; the flip of the edge bc would replace these two triangles by the triangles abd and adc. The flip operation is only considered when it does not produce a self-intersecting surface. In this paper we show that given two surfaces S1, [Formula: see text], it is possible that there is no sequence of flips transforming S1 into S2, even in the case that V consists of points in convex position.

scholarly journals Drawing Hamiltonian Cycles with no Large Angles

10.37236/2356 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Adrian Dumitrescu ◽  
János Pach ◽  
Géza Tóth
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Point Sets ◽  
Convex Position ◽  
Straight Line ◽  
Open Region ◽  
Point Set ◽  
Rectifiable Curves ◽  
Element Set

Let $n \geq 4$ be even. It is shown that every set $S$ of $n$ points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of $n$ straight-line edges such that the angle between any two consecutive edges is at most $2\pi/3$. For $n=4$ and $6$, this statement is tight. It is also shown that every even-element point set $S$ can be partitioned  into at most two subsets, $S_1$ and $S_2$, each admitting a spanning tour with no angle larger than $\pi/2$. Fekete and Woeginger conjectured that for sufficiently large even $n$, every $n$-element set admits such a spanning tour. We confirm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by finitely many rectifiable curves: for any $\epsilon>0$, these sets almost surely admit a spanning tour with no angle larger than $\epsilon$.

A modular version of the Erdős– Szekeres theorem

2001 ◽  
Vol 38 (1-4) ◽  
pp. 245-260 ◽  
Author(s):  
Gy. Károlyi ◽  
J. Pach ◽  
G. Tóth
Keyword(s):  
Convex Hull ◽  
General Position ◽  
Empty Convex ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Mod P

Bialostocki, Dierker, and Voxman proved that for any n = p +2, there is an integer B(n; p) with the following property. Every set of B(n; p) points in general position in the plane has n points in convex position such that the number of points in the interior of their convex hull is 0 mod p. They conjectured that the same is true for all pairs n = 3, p =2. In this note, we show that every su&ciently large point set determining no triangle with more than one point in its interior has n elements that form the vertex set of an empty convex n-gon. As a consequence, we show that the above conjecture is true for all n =5p=6+O(1).

scholarly journals More Turán-Type Theorems for Triangles in Convex Point Sets

10.37236/7224 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Boris Aronov ◽  
Vida Dujmović ◽  
Pat Morin ◽  
Aurélien Ooms ◽  
Luı́s Fernando Schultz Xavier da Silveira
Keyword(s):  
Packing Problem ◽  
Point Sets ◽  
Convex Position ◽  
Point Set ◽  
Forbidden Configurations ◽  
Convex Point

 We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations.  As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact.  This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

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
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