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Gülsüm Yeliz ŞENTÜRK, Salim YÜCE
THE DENSITIES OF THE SETS OF (NON) COLLINEAR POINTS AND INTERSECTING LINES IN THE EUCLIDEAN PLANE
 
Introduction: Between the years 1935 and 1939, Wilhelm Blaschke made a series of presentations entitled 'Integral Geometry' at the Mathematics Seminar at the University of Hamburg. Then, W. Blaschke (1949) published his this work in book entitled "Vorlesungen über Integralgeometrie". Indubitably, his studies have been accepted as the main references on integral geometry. The density for sets of points and also the density for sets of lines are defined and the geometric meanings of the integrals of these differential forms are given in that book. After his contributions, numerous studies have been examined on integral geometry in Euclidean and non-Euclidean spaces from past to the present. In order to apply the idea of probability to geometric objects, it is necessary to define a measure for a set of elements (such as points, lines, geodesics ..). Let x and y denote rectangular cartesian coordinates in the Euclidean plane. In integral geometry and in the theory of geometrical probability, the measure of a set of points in the Euclidean plane is defined by the integral, over the set, of the differential form $dX=dx\wedge dy$ which is called the density for points and the measure of a set of lines $x\cos \psi +y\sin \psi =h$ (i.e.$l(h,\psi )$) in the Euclidean plane is defined by the integral, over the set, of the differential form $dl=(dx\cos \psi +dy\sin \psi )\wedge d\psi ,$ which is called the density for lines, [1]. The density of sets of lines is independent of the point selection in the Euclidean plane, [1]. Luis A. Santaló (2004) referred both classical and modern Euclidean (non-Euclidean) integral geometry. Let $X({{x}_{1}},{{y}_{1}}),$$X(x_1,y_1),Y(x_2,y_2)$ be two sets of points in the Euclidean plane and $l( h, \psi )$ be a line passing through these points. The density of a pair of points is defined by the equation\[dX \wedge dY = d{x_1} \wedge d{y_1} \wedge d{x_2} \wedge d{y_2}\]. As a kind of dual problem of the density of a pair of points, the density of a pair of intersecting lines can be expressed, [2]. Purpose: The purpose of this study is to give essential properties of the densities of sets of collinear points, non collinear points and intersecting lines. Scope: We explain the connection among the densities of the sets of collinear points, among the densities of the sets of non collinear points, and among the densities of the sets of intersecting lines in the Euclidean plane. Limitations: The study is limited to Euclidean plane. Moreover, the points are not moving point along the coordinate axes and along the lines . Method: As a beginning, we shortly present formulas of the density of a set of points, the density of a set of lines, the density for a pair of points, the density for a pair of intersecting lines in the Euclidean plane, respectively. And then, we study on collinear points, non collinear points and intersecting lines in the Euclidean plane. Findings: By means of these density formulas, we get some density properties of the set of collinear, non collinear points and lines in the Euclidean plane. Conclusion: In this study, we give the relationship among the densities of sets of collinear points, among the densities of sets of non collinear points and among the densities of sets of intersecting lines in the Euclidean plane.

Anahtar Kelimeler: Line, Point, Density



 


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