Number Of Empty Triangles Research Articles

No selection lemma for empty triangles

Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of Θ(n3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^3)$$\\end{document} triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them.For families of Θ(n3-α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^{3-\\alpha})$$\\end{document} triangles, with 0≤α≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\alpha \\le 1$$\\end{document}, there might not be a point in more than Θ(n3-2α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^{3-2\\alpha})$$\\end{document} of those triangles.An empty triangle of P is a triangle spanned by Pnot containing any point of P in its interior. Bárány conjectured that there exists an edgespanned by P that is incident to a super-constant number of empty triangles of P. The number of empty trianglesof P might be as low as Θ(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^2)$$\\end{document}; in such a case, on average, every edge spanned by P is incident to a constant numberof empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper boundmight not hold. In this paper we show that, somewhat surprisingly,the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number 0≤α≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\leq \\alpha \\leq 1$$\\end{document} there exists a point set of size n with Θ(n3-α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta(n^{3-\\alpha})$$\\end{document} empty triangles such that any point of the plane is only in O(n3-2α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^{3-2\\alpha})$$\\end{document} empty triangles.

Long Term Results of Visual Field Progression Analysis in Open Angle Glaucoma Patients Under Treatment.

Purpose : To evaluate visual field progression with trend and event analysis in open angle glaucoma patients under treatment.Materials and Methods : Fifteen year follow-up results of 408 eyes of 217 glaucoma patients who were followed at Adnan Menderes University, Department of Ophthalmology between 1998 and 2013 were analyzed retrospectively. Visual field data were collected for Mean Deviation (MD), Visual Field Index (VFI), and event occurrence.Results : There were 146 primary open-angle glaucoma (POAG), 123 pseudoexfoliative glaucoma (XFG) and 139 normal tension glaucoma (NTG) eyes. MD showed significant change in all diagnostic groups (p<0.001). The difference of VFI between first and last examinations were significantly different in POAG (p<0.001), and XFG (p<0.003) but not in NTG. VFI progression rates were -0.3, -0.43, and -0.2 % loss/year in treated POAG, XFG, and NTG, respectively. The number of empty triangles were statistically different between POAG-NTG (p=0.001), and XFG-NTG (p=0.002) groups. The number of half-filled (p=0.002), and full-filled (p=0.010) triangles were significantly different between XFG-NTG groups.Conclusion : Functional long-term follow-up of glaucoma patients can be monitored with visual field indices. We herein report our fifteen year follow-up results in open angle glaucoma.

Empty Triangles in Complete Topological Graphs

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. Let $$G$$G be a complete simple topological graph on $$n$$n vertices. The three edges induced by any triplet of vertices in $$G$$G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth proved that $$G$$G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least $$2n/3$$2n/3. We settle Harborth's conjecture in the affirmative.

Empty Triangles in Good Drawings of the Complete Graph

A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise connected vertices form a Jordan curve which we call a triangle. We say that a triangle is empty if one of the two connected components it induces does not contain any of the remaining vertices of the drawing of the graph. We show that the number of empty triangles in any good drawing of the complete graph \(K_n\) with \(n\) vertices is at least \(n\).

Empty triangles in complete topological graphs

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. Let G be a simple complete topological graph. The three edges induced by any triple of vertices in G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth [Harborth, Heiko, Empty triangles in drawings of the complete graph, Discrete Mathematics, 191 (1998), 109–111; Brass, Peter, William O. J. Moser and János Pach, “Research Problems in Discrete Geometry” Springer, (2005)] proved that G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least 2n/3. In this note, we verify Harborthʼs conjecture.

Many Empty Triangles have a Common Edge

Given a finite point set $$X$$ in the plane, the degree of a pair $$\{x,y\} \subset X$$ is the number of empty triangles $$t=\mathrm {conv} \{x,y,z\},$$ where empty means $$t\cap X=\{x,y,z\}.$$ Define $$\deg X$$ as the maximal degree of a pair in $$X.$$ Our main result is that if $$X$$ is a random sample of $$n$$ independent and uniform points from a fixed convex body, then $$\deg X \ge cn/\ln n$$ in expectation.

Finding minimum area simple pentagons

Given a set P of n points in the plane, we want to find a simple, not necessarily convex, pentagon Q with vertices in P of minimum area. We present an algorithm for solving this problem in time O( nT( n)) and space O( n), where T( n) is the number of empty triangles in the set.