scholarly journals Sparse Distance Sets in the Triangular Lattice

10.37236/3263 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Tanbir Ahmed ◽  
Hunter Snevily
Keyword(s):  
Triangular Lattice ◽  
Euclidean Plane ◽  
Optimal Point ◽  
Planar Point ◽  
Point Configurations ◽  
Point Subset ◽  
Point Set ◽  
Distance Set ◽  
Distance Sets ◽  
Increasing Function

A planar point-set $X$ in Euclidean plane is called a $k$-distance set if there are exactly $k$ different distances among the points in $X$. The function $g(k)$ denotes the maximum number of points in the Euclidean plane that is a $k$-distance set. In 1996, Erdős and Fishburn conjectured that for $k\geq 7$, every $g(k)$-point subset of the plane that determines $k$ different distances is similar to a subset of the triangular lattice. We believe that if $g(k)$ is an increasing function of $k$, then the conjecture is false. We present data that supports our claim and a method of construction that unifies known optimal point configurations for $k\geq 3$.

scholarly journals A Proof of Erdős-Fishburn's Conjecture for $g(6)=13$

10.37236/2917 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Wei Xianglin
Keyword(s):  
Euclidean Plane ◽  
Interesting Problem ◽  
Planar Point ◽  
Point Set ◽  
Distance Set

A planar point set $X$ in the Euclidean plane is called a $k$-distance set if there are exactly $k$ distances between two distinct points in $X$. An interesting problem is to find the largest possible cardinality of a $k$-distance set. This problem was introduced by Erdős and Fishburn (1996). Maximum planar sets that determine $k$ distances for $k$ less than 5 have been identified. The 6-distance conjecture of Erdős and Fishburn states that 13 is the maximum number of points in the plane that determine exactly 6 different distances. In this paper, we prove the conjecture.

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.

BASELINE BOUNDED HALF-PLANE VORONOI DIAGRAM

2013 ◽  
Vol 05 (03) ◽  
pp. 1350021 ◽  
Author(s):  
BING SU ◽  
YINFENG XU ◽  
BINHAI ZHU
Keyword(s):  
Voronoi Diagram ◽  
Half Plane ◽  
Line Segment ◽  
Euclidean Plane ◽  
Voronoi Region ◽  
Point Set ◽  
Set Of Points

Given a set of points P = {p1, p2, …, pn} in the Euclidean plane, with each point piassociated with a given direction vi∈ V. P(pi, vi) defines a half-plane and L(pi, vi) denotes the baseline that is perpendicular to viand passing through pi. Define a region dominated by piand vias a Baseline Bounded Half-Plane Voronoi Region, denoted as V or(pi, vi), if a point x ∈ V or(pi, vi), then (1) x ∈ P(pi, vi); (2) the line segment l(x, pi) does not cross any baseline; (3) if there is a point pj, such that x ∈ P(pj, vj), and the line segment l(x, pj) does not cross any baseline then d(x, pi) ≤ d(x, pj), j ≠ i. The Baseline Bounded Half-Plane Voronoi Diagram, denoted as V or(P, V), is the union of all V or(pi, vi). We show that V or(pi, vi) and V or(P, V) can be computed in O(n log n) and O(n2log n) time, respectively. For the heterogeneous point set, the same problem is also considered.

scholarly journals Heronian Friezes

2020 ◽  
Author(s):  
Sergey Fomin ◽  
Linus Setiabrata
Keyword(s):  
Computational Geometry ◽  
Euclidean Plane ◽  
Type A ◽  
Cluster Algebras ◽  
Algebraic Condition ◽  
Related Family ◽  
Point Configurations ◽  
Laurent Phenomenon ◽  
Geometric Origin ◽  
Propagation Rule

Abstract Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type $A$, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter’s frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic) and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley–Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley–Menger friezes, as well as the corresponding periodicity results.

scholarly journals On the Oβ-hull of a planar point set

2018 ◽  
Vol 68 ◽  
pp. 277-291 ◽  
Author(s):  
Carlos Alegría-Galicia ◽  
David Orden ◽  
Carlos Seara ◽  
Jorge Urrutia
Keyword(s):  
Planar Point ◽  
Point Set
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