Distinct Triangle Areas in a Planar Point Set

2007 ◽  
pp. 119-129 ◽  
Author(s):  
Adrian Dumitrescu ◽  
Csaba D. Tóth
Keyword(s):  
Planar Point ◽  
Point Set ◽  
Distinct Triangle

scholarly journals Distinct Triangle Areas in a Planar Point Set over Finite Fields

10.37236/700 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Finite Plane ◽  
Expected Number ◽  
Planar Point ◽  
Point Set ◽  
Distinct Triangle

Let $\mathcal{P}$ be a set of $n$ points in the finite plane $\mathbb{F}_q^2$ over the finite field $\mathbb{F}_q$ of $q$ elements, where $q$ is an odd prime power. For any $s \in \mathbb{F}_q$, denote by $A (\mathcal{P}; s)$ the number of ordered triangles whose vertices in $\mathcal{P}$ having area $s$. We show that if the cardinality of $\mathcal{P}$ is large enough then $A (\mathcal{P}; s)$ is close to the expected number $|\mathcal{P}|^3/q$.

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 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

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.

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