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.

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 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 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 .

PARAMETRIC POLYMATROID OPTIMIZATION AND ITS GEOMETRIC APPLICATIONS

2002 ◽  
Vol 12 (05) ◽  
pp. 429-443 ◽  
Author(s):  
NAOKI KATOH ◽  
HISAO TAMAKI ◽  
TAKESHI TOKUYAMA
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Transportation Problem ◽  
Minimum Weight ◽  
Planar Point ◽  
Line Arrangement ◽  
Optimal Bound ◽  
Straight Line ◽  
Point Set ◽  
Multiple Levels

We give an optimal bound on the number of transitions of the minimum weight base of an integer valued parametric polymatroid. This generalizes and unifies Tamal Dey's O(k1/3 n) upper bound on the number of k-sets (and the complexity of the k-level of a straight-line arrangement), David Eppstein's lower bound on the number of transitions of the minimum weight base of a parametric matroid, and also the Θ(kn) bound on the complexity of the at-most-k level (the union of i-levels for i = 1,2,…,k) of a straight-line arrangement. As applications, we improve Welzl's upper bound on the sum of the complexities of multiple levels, and apply this bound to the number of different equal-sized-bucketings of a planar point set with parallel partition lines. We also consider an application to a special parametric transportation problem.

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 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.

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.

scholarly journals Counting Plane Graphs: Cross-Graph Charging Schemes

2013 ◽  
Vol 22 (6) ◽  
pp. 935-954 ◽  
Author(s):  
MICHA SHARIR ◽  
ADAM SHEFFER
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Upper Bounds ◽  
Straight Edge ◽  
Specific Point ◽  
Plane Graphs ◽  
Fixed Set ◽  
Point Set ◽  
Vertex Degrees ◽  
Set Of Points

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have recently been used to obtain various properties of triangulations that are embedded in a fixed set of points in the plane. We generalize this method to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound ofO*(187.53N) (where theO*(⋅) notation hides polynomial factors) for the maximum number of crossing-free straight-edge graphs that can be embedded in any specific set ofNpoints in the plane (improving upon the previous best upper bound 207.85Nin Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on the expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded in a specific point set.We then apply the cross-graph charging-scheme method to graphs that allow certain types of crossings. Specifically, we consider graphs with no set ofkpairwise crossing edges (more commonly known ask-quasi-planar graphs). Fork=3 andk=4, we prove that, for any setSofNpoints in the plane, the number of graphs that have a straight-edgek-quasi-planar embedding overSis only exponential inN.

Chebychev Approximations of Finite Point Sets with Application to Planar Kinematic Synthesis

1979 ◽  
Vol 101 (1) ◽  
pp. 32-40 ◽  
Author(s):  
Y. L. Sarkisyan ◽  
K. C. Gupta ◽  
B. Roth
Keyword(s):  
Point Sets ◽  
Finite Point ◽  
Kinematic Synthesis ◽  
Radial Deviation ◽  
Point Set ◽  
Planar Linkages ◽  
Straight Lines ◽  
Set Of Points

In the first part of this paper we consider the problem of determining circles which best approximate a given set of points. The approximation is one which minimizes the maximum radial deviation of the points from the approximating circle. Then a similar procedure is developed for determining straight lines which best approximate a given point set. The final parts of the paper illustrate the application of these results to synthesizing planar linkages.

FITTING FLATS TO POINTS WITH OUTLIERS

2011 ◽  
Vol 21 (05) ◽  
pp. 559-569
Author(s):  
GUILHERME D. DA FONSECA
Keyword(s):  
Lower Bound ◽  
Computer Science ◽  
Upper Bound ◽  
Arbitrary Constant ◽  
Fundamental Problem ◽  
Constant Factor ◽  
Infinite Cylinder ◽  
Running Time ◽  
Point Set ◽  
Set Of Points

Determining the best shape to fit a set of points is a fundamental problem in many areas of computer science. We present an algorithm to approximate the k-flat that best fits a set of n points with n - m outliers. This problem generalizes the smallest m-enclosing ball, infinite cylinder, and slab. Our algorithm gives an arbitrary constant factor approximation in O(nk+2/m) time, regardless of the dimension of the point set. While our upper bound nearly matches the lower bound, the algorithm may not be feasible for large values of k. Fortunately, for some practical sets of inliers, we reduce the running time to O(nk+2/mk+1), which is linear when m = Ω(n).

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