scholarly journals Note on Gauss' Proof of the Reciprocity of Parallelism

1913 ◽  
Vol 32 ◽  
pp. 15-18
Author(s):  
D. M. Y. Sommerville
Keyword(s):  
Unique Line ◽  
Definition Of ◽  
Straight Lines

The proposition that if AA′‖BB′ then BB′‖AA′ appears at first sight so simple that it might be regarded as almost intuitive. This is because we already think of parallelism as a symmetrical relationship between two straight lines, in accordance with Euclid's definition of parallels as “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” If we take along with this definition Euclid's fifth postulate, or Playfair's equivalent, it defines a unique line through a given point parallel to a given line; but, without the postulate, it cannot be assumed to define more than a class of lines, and a stricter definition is required.

History of Air Maps and Charts

1998 ◽  
Vol 51 (2) ◽  
pp. 203-215 ◽  
Author(s):  
Philip Steele
Keyword(s):  
Nineteenth Century ◽  
Ordnance Survey ◽  
History Of ◽  
Minor Significance ◽  
Aerial Navigation ◽  
The Difference ◽  
Definition Of ◽  
Straight Lines ◽  
The Hill

There is no generally accepted definition of the difference between a map and a chart. A widespread feeling probably exists favouring the old saying that maps are to look at and charts to work on. It is true that the term ‘aeronautical chart’ gained a general currency over alternative terms as contact flying gave way to aerial navigation. But, in this paper, the terms ‘map’ and ‘chart’ will be used as seems appropriate to each occasion, without attempt to conform to any particular definition.We can get an idea of what was available to the earliest aviators by looking at an Ordnance Survey reprint of one of their nineteenth century maps (Fig. 1). They are printed in one colour only, black on white. By far the predominant feature is the hill shading. Quite gentle hills are hachured with a heaviness which tends to obscure both natural features like rivers, lakes and woodlands and man-made constructions such as towns and villages, roads, canals and railways. Hills are, of course, very important features to those on the ground, since they limit the extent to which other features can be seen. To the soldier, the significance of high ground is self-evident, and it was principally for the ordnance requirements of soldiers that these maps had been developed. But when men began to view the ground from the air, the perspective changed. Hills appeared flattened out and, provided that you knew the height of the tallest in the area and were sure none would impede your take-off or landing, were of minor significance. Lakes and woods, though, were spread out before you in their distinctive shapes, while railway lines and canals presented bold straight lines and curves, and rivers their unique courses, to your view. The need was for new kinds of maps which would give due prominence to such features.

scholarly journals On a conjecture concerning helly circle graphs

2003 ◽  
Vol 23 (1) ◽  
pp. 221-229 ◽  
Author(s):  
Guillermo Durán ◽  
Agustín Gravano ◽  
Marina Groshaus ◽  
Fábio Protti ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Systems Of Equations ◽  
Intersection Graphs ◽  
Circle Graph ◽  
Circle Graphs ◽  
Analytic Tool ◽  
Two Systems ◽  
Cartesian Plane ◽  
Definition Of ◽  
Straight Lines

We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle), and thus this method provides an analytic way for recognizing circle graphs. A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. A conjecture by Durán (2000) states that G is a Helly circle graph if and only if G is a circle graph and contains no induced diamonds (a diamond is a graph formed by four vertices and five edges). Many unsuccessful efforts - mainly based on combinatorial and geometrical approaches - have been done in order to validate this conjecture. In this work, we utilize the ideas behind the definition of e-circle graphs and restate this conjecture in terms of an equivalence between two systems of equations and inequations, providing a new, analytic tool to deal with it.

Graphic-Analytical Researches of Involution

2017 ◽  
Vol 5 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Графский ◽  
O. Grafskiy ◽  
Усманов ◽  
A. Usmanov ◽  
Холодилов ◽  
...  
Keyword(s):  
Cross Ratio ◽  
Rectangular Coordinate System ◽  
Projective Transformations ◽  
Coordinate Axis ◽  
Graphic Editor ◽  
Applied Informatics ◽  
The Cross ◽  
Pass Through ◽  
Definition Of ◽  
Straight Lines

Known projective transformations, namely their private types such as harmonism and involution are considered. It is known that projective transformations are collinear, at their performance the order, the cross ratio of fours of elements (on a straight line — the cross ratio of four points, in a bunch of straight lines — the cross ratio of this bunch’s four straight lines, this property (invariant) is similarly preserved for a bunch of planes, i.e. in considering of first step forms) is preserved. At a constructive approach to such transformations there are some ways for definition of position for corresponding elements which students use when studying discipline "Affine and projective geometry" on preparation profiles 09.03.01 — “CAD Systems" and 09.03.03 — “Applied Informatics in Design”. The received constructions are checked by analytical calculations, proceeding from known dependences for harmonism and involutions. In such a case results both for a range of points, and for a bunch of straight lines which pass through these points are analytically compared. The provided computational and graphic work contains three sections: prospects, harmonism and involution, and is carried out by students on individual options with application of the graphic editor Microsoft Visio or the graphic package CO MPAS voluntary. In the present paper some constructions in definition of corresponding points in elliptic and hyperbolic involution are considered, some of these constructions are published for the first time. Besides, a proposition has been formulated: in a rectangular coordinate system the work for coordinates of two points related to a circle intersection with one coordinate axis is equal to the product for coordinates of two other points related to this circle intersection with the other coordinate axis. This proposition is fairly for imaginary points of circle intersection with coordinate axes as well.

Shape Features Detection Based on Hough Transform in Images

2014 ◽  
Vol 644-650 ◽  
pp. 1104-1106 ◽  
Author(s):  
Guang Li Chu ◽  
Yan Jie Wang
Keyword(s):  
Hough Transform ◽  
Template Matching ◽  
Traditional Method ◽  
Detection Method ◽  
Feature Points ◽  
Shape Features ◽  
Line Segments ◽  
Definition Of ◽  
Straight Lines ◽  
And Storage

Hough transform as an effective graphics target detection method can detect straight lines, circles, ellipses, parabolas and many other analytical graphics. The discretization of space, as well as the calculation of the process make Hough transform have some limitations, such as poor detection results because of high-intensity noise, a large amount of calculation, large demand of storage resources and so on. This paper analyzes the Hough Transform voting process and points out that the accumulation with 1 in the method is unreasonable. The paper proposed a Hough transform based on template matching via the modification of the definition of the traditional method. In this method, each parameter unit identifies a template in image space. The feature points according with the conditions can be searched by the template actively. The method takes the number of feature points as the value of parameter unit and takes the record of the coordinates of line segment endpoints. So line segments can be detected and storage resources can be saved.

scholarly journals Worm Selector: Volume Selection in a 3D Point Cloud Through Adaptive Modelling

2017 ◽  
Vol 17 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Emmanuel Dubois ◽  
Adrien Hamelin
Keyword(s):  
Point Cloud ◽  
User Study ◽  
Point Clouds ◽  
Ease Of Use ◽  
Free Form ◽  
3D Point Cloud ◽  
3D Point Clouds ◽  
3D Scanners ◽  
Definition Of ◽  
Straight Lines

3D point clouds are more and more widely used, especially because of the proliferation of manual and cheap 3D scanners and 3D printers. Due to the large size of the 3D point clouds, selecting part of them is very often required. Existing interaction techniques include ray/cone casting and predefined or free-form selection volume. In order to cope with the traditional trade-off between accuracy, ease of use and flexibility of these different forms of selection techniques in a 3D point cloud, we present the Worm Selector. It allows to select complex shapes while remaining simple to use and accurate. Using the Worm Selector relies on three principles: 1) points are selected by progressively constructing a cylinder-like shape (the adaptative worm) through the sequential definition of several sections; 2) a section is defined as a set of two contours linked together with straight lines; 3) each contour is a freely drawn closed shape. A user study reveals that the Worm Selector is significantly faster than a classical selection mechanism based on predefined volumes such as spheres or cuboids, while maintaining a comparable level of precision and recall.

Territorial Disputes and Their Resolution in the Recent Jurisprudence of the International Court of Justice

2017 ◽  
Vol 31 (1) ◽  
pp. 117-146 ◽  
Author(s):  
HUGH THIRLWAY
Keyword(s):  
Territorial Disputes ◽  
Territorial Dispute ◽  
International Court Of Justice ◽  
Court Of Justice ◽  
Boundary Disputes ◽  
International Court ◽  
Territorial Claims ◽  
Definition Of ◽  
Sovereign Territory ◽  
Straight Lines

AbstractThe workload of the International Court of Justice in recent years has increasingly featured cases of disputes classified either as ‘territorial disputes’ or as ‘boundary disputes’, or otherwise involving the Court in considerations of the law relating to acquisition or transmission of territory, or to the creation, location and effect of territorial frontiers. The present survey analyzes the contributions to international law of the Court's decisions in these recent cases. Matters examined include the significance of the terms ‘boundary dispute’ or ‘territorial dispute’; the definition of what constitutes sovereign territory; titles andeffectivitésas bases for territorial claims; decolonization and theuti possidetis juris; use of natural features or of straight lines as boundaries; and relations across a frontier once established.

scholarly journals III.—On the Extension of Brouncker's Method to the Comparison of Several Magnitudes.

1870 ◽  
Vol 26 (1) ◽  
pp. 59-67
Author(s):  
Edward Sang
Keyword(s):  
The Other ◽  
Exact Science ◽  
Straight Line ◽  
The One ◽  
Definition Of ◽  
Straight Lines

The discovery of those numbers which shall, either truly or approximately, represent the ratio of two magnitudes, necessarily attracted the attention of the earliest cultivators of exact science. The definition of the equality of ratios given in Euclid's compilation clearly exposes the nature of the process used in his time. This process consisted in repeating each of the two magnitudes until some multiple of the one agreed perfectly or nearly with a multiple of the other; the numbers of the repetitions, taken in inverse order, represented the ratio. Thus, if the proposed magnitudes were two straight lines, Euclid would have opened two pairs of compasses, one to each distance, and, beginning at some point in an indefinite straight line, he would step the two distances along, bringing up that which lagged behind, until he obtained an exact or a close coincidence.

Newton’s Bucket

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Fundamental Problem ◽  
Uniform Motion ◽  
Newtonian Mechanics ◽  
The Core ◽  
Straight Line ◽  
Definition Of ◽  
Straight Lines ◽  
Absolute Structure

This chapter describes the fundamental problem at the core of Newton’s dynamics: the definition of inertia. This is provided by an absolute structure in Newtonian mechanics, but, as Leibniz and later Mach argued, it should be dynamically determined. This is the core of Newton’s famous ‘bucket experiment’. Assuming this law as a postulate, without first defining the notions of ‘rest’, ‘uniform motion’ and ‘right (or straight) line’, is inconsistent. In a universe that is, in Barbour’s words, like ‘bees swarming in nothing’, how is one to talk about rest/uniform motion/straight lines? With respect to what? The problem is that of establishing a notion of equilocality: in an everchanging universe, what does it mean for an object to be at the same place at dierent times?

scholarly journals Set estimation under biconvexity restrictions

2020 ◽  
Vol 24 ◽  
pp. 770-788
Author(s):  
Alejandro Cholaquidis ◽  
Antonio Cuevas
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Optimization Theory ◽  
Minimal Set ◽  
Set Estimation ◽  
Convex Case ◽  
Natural Notion ◽  
Inclination Angles ◽  
Definition Of ◽  
Straight Lines

A set in the Euclidean plane is said to be biconvex if, for some angle θ ∈ [0, π∕2), all its sections along straight lines with inclination angles θ and θ + π∕2 are convex sets (i.e., empty sets or segments). Biconvexity is a natural notion with some useful applications in optimization theory. It has also be independently used, under the name of “rectilinear convexity”, in computational geometry. We are concerned here with the problem of asymptotically reconstructing (or estimating) a biconvex set S from a random sample of points drawn on S. By analogy with the classical convex case, one would like to define the “biconvex hull” of the sample points as a natural estimator for S. However, as previously pointed out by several authors, the notion of “hull” for a given set A (understood as the “minimal” set including A and having the required property) has no obvious, useful translation to the biconvex case. This is in sharp contrast with the well-known elementary definition of convex hull. Thus, we have selected the most commonly accepted notion of “biconvex hull” (often called “rectilinear convex hull”): we first provide additional motivations for this definition, proving some useful relations with other convexity-related notions. Then, we prove some results concerning the consistent approximation of a biconvex set S and the corresponding biconvex hull. An analogous result is also provided for the boundaries. A method to approximate, from a sample of points on S, the biconvexity angle θ is also given.

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