On One Property of a Circle on the Coordinate Plane

2017 ◽  
Vol 5 (2) ◽  
pp. 13-24
Author(s):  
Графский ◽  
O. Grafskiy ◽  
Пономарчук ◽  
Yu. Ponomarchuk
Keyword(s):  
Euclidean Geometry ◽  
Coordinate Plane ◽  
Non Euclidean Geometry ◽  
Geometrical Form ◽  
Constant Distance ◽  
Intersection Points ◽  
Definition Of ◽  
Set Of Points ◽  
First Time

Descartes’ and Fermat's method allowed to define many geometrical forms, including circles, on the coordinate plane by means of the arithmetic equations and to make necessary analytical operations in order to solve many problems of theoretical and applied research in various scientific areas, for example. However, the equations of a circle and other conics in the majority of research topics are used in the subsequent analysis of applied problems, or for analytical confirmation of constructive solutions in geometrical research, according to Russian geometrician G. Monge and others, including. It is natural to consider a circle as a locus of points, equidistant from a given point — a center of the circle, with a constant distance R. There is another definition of a circle: a set of points from which a given segment is visible under constant directed angle. Besides, a circle is accepted to model the Euclid plane in the known scheme of non-Euclidean geometry of Cayley-Klein, it is the absolute which was given by A. Cayley for the first time in his memoirs. It is possible to list various applications of this geometrical form, especially for harmonism definition of the corresponding points, where the diametral opposite points of a circle are accepted as basic, and also for construction of involutive compliances. The construction of tangents to a circle can be considered as a classical example. Their constructive definition is simple, but also constructions on the basis of known projective geometry postulates are possible (a hexagon when modeling a series of the second order, Pascal's lines). These postulates can be applied to construction of tangents to a circle (to an ellipse and hyperboles to determination of imaginary points of intersection of a circle and a line. This paper considers the construction of tangents to a circle without the use of arches of auxiliary circles, which was applied in order to determine the imaginary points of intersection of a circle and a line (an axis of coordinates). Besides, various dependences of parameter p2, which is equal to the product of the values of the intersection points’ coordinates of a circle and coordinate axes, are analytically determined.

scholarly journals KONSISTENSI AKSIOMA-AKSIOMA TERHADAP ISTILAH-ISTILAH TAKTERDEFINISI GEOMETRI HIPERBOLIK PADA MODEL PIRINGAN POINCARE

EDUPEDIA ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 161
Author(s):  
Febriyana Putra Pratama ◽  
Julan Hernadi
Keyword(s):  
Half Plane ◽  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Cross Ratio ◽  
Literature Study ◽  
Disk Model ◽  
Non Euclidean Geometry ◽  
The Cross ◽  
Definition Of ◽  
Poincaré Disk

This research aims to know the interpretation the undefined terms on Hyperbolic geometry and it’s consistence with respect to own axioms of Poincare disk model. This research is a literature study that discusses about Hyperbolic geometry. This study refers to books of Foundation of Geometry second edition by Gerard A. Venema (2012), Euclidean and Non Euclidean Geometry (Development and History)  by Greenberg (1994), Geometry : Euclid and Beyond by Hartshorne (2000) and Euclidean Geometry: A First Course by M. Solomonovich (2010). The steps taken in the study are: (1) reviewing the various references on the topic of Hyperbolic geometry. (2) representing the definitions and theorems on which the Hyperbolic geometry is based. (3) prepare all materials that have been collected in coherence to facilitate the reader in understanding it. This research succeeded in interpret the undefined terms of Hyperbolic geometry on Poincare disk model. The point is coincide point in the Euclid on circle . Then the point onl γ is not an Euclid point. That point interprets the point on infinity. Lines are categoried in two types. The first type is any open diameters of   . The second type is any open arcs of circle. Half-plane in Poincare disk model is formed by Poincare line which divides Poincare field into two parts. The angle in this model is interpreted the same as the angle in Euclid geometry. The distance is interpreted in Poincare disk model defined by the cross-ratio as follows. The definition of distance from  to  is , where  is cross-ratio defined by  . Finally the study also is able to show that axioms of Hyperbolic geometry on the Poincare disk model consistent with respect to associated undefined terms.

DETERMINATION OF ECONOMIC DAMAGE FROM DETERIORATION/DESTRUCTION OF ECOSYSTEM SERVICES

2018 ◽  
pp. 43-48
Author(s):  
Oksana Veklych
Keyword(s):  
Ecosystem Services ◽  
Cost Estimation ◽  
Economic Loss ◽  
Economic Damage ◽  
Structural Scheme ◽  
Definition Of ◽  
The Cost ◽  
First Time

The definition of economic damage from the deterioration/destruction of ecosystem services and analytical structuring of the economic loss from it were given for the first time. It was proposed and disclosed the logic-structural scheme that describes the algorithm of the sequence of actions and calculations for carrying out the cost estimation of damage from deterioration/destruction of ecosystem services in order to further substantiate the recommendations for additional filling of local budgets and attraction of targeted investments for implementation of projects aimed at conservation and restoration ecosystems.

PRECISE DETERMINATION OF BACKBONE STRUCTURE AND CONDUCTIVITY OF 3D PERCOLATION NETWORKS BY THE DIRECT ELECTRIFYING ALGORITHM

2009 ◽  
Vol 20 (03) ◽  
pp. 423-433 ◽  
Author(s):  
CHUNYU LI ◽  
TSU-WEI CHOU
Keyword(s):  
Efficient Algorithm ◽  
Precise Determination ◽  
Scaling Exponents ◽  
Resistor Network ◽  
Backbone Structure ◽  
Definition Of ◽  
First Time ◽  
Good Agreement

This paper confirms the applicability of a newly developed efficient algorithm, the direct electrifying method, for identifying backbone for 3D site and bond percolating networks. This algorithm is based on the current-carrying definition of backbone and carried out on the predetermined spanning cluster, which is assumed to be a resistor network. The scaling exponents so obtained for backbone mass, red bonds, and conductivity are in very good agreement with some existing results. The perfectly balanced bonds in 3D backbone structures are predicted first time to be 0.00179 ± 0.00009 and 0.00604 ± 0.00008 of the backbone mass for bond and site percolations, respectively.

scholarly journals The Topological Reading of Ambiances in the Built Environment: The New Methodology for the Analysis of the Luminous Ambiance in the Museum Space

2019 ◽  
Vol 64 ◽  
pp. 03004
Author(s):  
Selma Saraoui ◽  
Azeddine Belakehal ◽  
Abdelghani Attar ◽  
Amar Bennadji
Keyword(s):  
Sequential Analysis ◽  
Euclidean Geometry ◽  
Essential Element ◽  
Analysis Model ◽  
Architectural Space ◽  
Spatial Features ◽  
Non Euclidean Geometry ◽  
Metric Dimensions ◽  
Definition Of ◽  
Shed Light

Daylight is currently at the centre of discourse on architectural space. The definition of architectural space takes essence from Euclidean geometry related to metric dimensions. The present study is an attempt to shed light on topology which is a non-Euclidean geometry. It can support non-metric components of space such as light to define architectural space. A corpus of six European museums has been chosen to study the immaterial or material relationships between form and daylight, since light is an essential element for the success of the exhibition. It also seeks to highlight discontinuity reports, and to confirm their existence through their software visualizations Therefore, the current study has taken into account an analysis model based on the notions of "route" and "sequence". The contemporary architectural project focused on taking into account human postures, both physical and psychological, within the architectural space. The results obtained show that light can release other spatial features for the museum space that can be highlighted by visualization with sequential analysis.

scholarly journals XIV. On mathematical concepts of the material world

1906 ◽  
Vol 205 (387-401) ◽  
pp. 465-525 ◽  
Keyword(s):  
Physical Science ◽  
Euclidean Geometry ◽  
Ordinary Language ◽  
Abstract Form ◽  
Mathematical Concepts ◽  
Material World ◽  
Mathematical Investigation ◽  
Definition Of ◽  
Physical Laws ◽  
Set Of Points

The object of this memoir is to initiate the mathematical investigation of various possible ways of conceiving the nature of the material world. In so far as its results are worked out in precise mathematical detail, the memoir is concerned with the possible relations to space of the ultimate entities which (in ordinary language) constitute the “stuff” in space. An abstract logical statement of this limited problem, in the form in which it is here conceived, is as follows: Given a set of entities which form the field of a certain polyadic ( i. e ., many-termed) relation R, what “axioms” satisfied by R have as their consequence, that the theorems of Euclidean geometry are the expression of certain properties of the field of R ? If the set of entities are themselves to be the set of points of the Euclidean space, the problem, thus set, narrows itself down to the problem of the axioms of Euclidean geometry. The solution of this narrower problem of the axioms of geometry is, assumed ( cf . Part II., Concept I.) without proof in the form most convenient for this wider investigation. But in Concepts III., IV., and V., the entities forming the field of R are the “stuff,” or part of the “stuff,” constituting the moving material world. Poincaré has used language which might imply the belief that, with the proper definitions, Euclidean geometry can be applied to express properties of the field of any polyadic relation whatever. His context, however, suggests that his thesis is, that in a certain sense (obvious to mathematicians) the Euclidean and certain other geometries are interchangeable, so that, if one can be applied, then each of the others can also be applied. Be that as it may, the problem, here discussed, is to find various formulations of axioms concerning R, from which, with appropriate definitions, the Euclidean geometry issues as expressing properties of the field of R. In view of the existence of change in the material world, the investigation has to be so conducted as to introduce, in its abstract form, the idea of time, and to provide for the definition of velocity and acceleration. The general problem is here discussed purely for the sake of its logical ( i. e ., mathematical) interest. It has an indirect bearing on philosophy by disentangling the essentials of the idea of a material world from the accidents of one particular concept. The problem might, in the future, have a direct bearing upon physical science if a concept widely different from the prevailing concept could be elaborated, which allowed of a simpler enunciation of physical laws. But in physical research so much depends upon a trained imaginative intuition, that it seems most unlikely that existing physicists would, in general, gain any advantage from deserting familiar habits of thought.

scholarly journals On mathematical concepts of the material world

1906 ◽  
Vol 77 (517) ◽  
pp. 290-291 ◽  
Keyword(s):  
Euclidean Space ◽  
Euclidean Geometry ◽  
Ordinary Language ◽  
Abstract Form ◽  
Mathematical Concepts ◽  
Material World ◽  
Mathematical Investigation ◽  
Definition Of ◽  
Set Of Points

The object of this memoir is to initiate the mathematical investigation of various possible ways of conceiving the nature of the Material World. In so far as its results are worked out in precise mathematical detail, the memoir is concerned with the possible relations to space of the ultimate entities which (in ordinary language) constitute the “stuff” in space. An abstract logical statement of this limited problem, in the form in which it is here conceived, is as follows:—Given a set of entities which form the field of a certain polyadic ( i. e ., many-termed) relation R . What “axioms” satisfied by R have as their consequence that the theorems of Euclidean Geometry are the expression of certain properties of the field of R ? If the set of entities are themselves to be the set of points of the Euclidean Space, the problem, thus set, narrows itself down to the problem of the axioms of Euclidean Geometry. The solution of this narrower problem of the axioms of geometry is assumed ( cf . Part II, Concept I) without proof in the form most convenient for this wider investigation. Poincaré has used language which might imply the belief that, with the proper definitions, Euclidean Geometry can be applied to express properties of the field of any polyadic relation whatever. His context, however, suggests that his thesis is, that in a certain sense (obvious to mathematicians) the Euclidean and certain other geometries are interchangeable, so that, if one can be applied, then each of the others can also be applied. Be that as it may, the problem here discussed is to find various formulations of axioms concerning R , from which, with appropriate definitions, the Euclidean Geometry issues as expressing properties of the field of R . In view of the existence of change in the Material World, the investigation has to be so conducted as to introduce, in its abstract form, the idea of time, and to provide for the definition of velocity and acceleration.

A new absolute determination of the acceleration due to gravity at the National Physical Laboratory, England

1967 ◽  
Vol 261 (1120) ◽  
pp. 211-252 ◽  
Keyword(s):  
National Physical Laboratory ◽  
Atomic Unit ◽  
Time Intervals ◽  
Gravity Station ◽  
Absolute Determination ◽  
Physical Laboratory ◽  
Air Resistance ◽  
Definition Of ◽  
First Time

A new absolute determination of the acceleration due to gravity at the National Physical Laboratory has been made by timing the symmetrical free motion of a body moving under the attraction of gravity; it is the first time this method has been used. The moving body was a glass ball and it was timed at its passage across two horizontal planes by the flashes of light that it produced when it passed between two horizontal slits which served to define each plane optically, the ball focusing light from one of the slits, which was illuminated, upon the other slit which had a photomultiplier placed behind it. The separation of the two planes defined by the pairs of slits was measured interferometrically and referred directly to the international wavelength definition of the metre, while the time intervals were measured in terms of the atomic unit of time scale A l. The value of gravity as reduced to the British Fundamental Gravity Station in the N. P. L. is 981 181.75 mgal, s.d. 0.13 mgal (1 mgal = 10 -5 m/s 2 ). Systematic errors, are believed to be very small; this is particularly true of the error due to air resistance. The main contribution to the observed scatter of the results comes from microseismic disturbances. The new result is 1.4 mgal less than that obtained at the fundamental station by J. S. Clark (1939) using a reversible pendulum . It is very close to the mean of a number of recent absolute determinations by other methods, but this may not be very significant because the uncertainties of those determinations and of the comparisons between the sites at which they were made and the present site are not less than 5 times the standard deviation of the new result.

Aristotle and the Consequentia Mirabilis

1957 ◽  
Vol 77 (1) ◽  
pp. 62-66 ◽  
Author(s):  
William Kneale
Keyword(s):  
Seventeenth Century ◽  
Euclidean Geometry ◽  
Early History ◽  
Philosophical Literature ◽  
Non Euclidean Geometry ◽  
History Of ◽  
First Time

In a passage of his Protrepticus mentioned by several ancient authors Aristotle wrote: εἰ μὲνφιλοσοφητέον φιλοσοφητέον, καὶ εἰ μὴ φιλοσοφητέον φιλοσοφητέον πάντως ἄρα φιλοσοφητέον (V. Rose, Aristotelis Fragmenta, 51. Cf. R. Walzer, Aristotelis Dialogorum Fragmenta, p. 22; W. D. Ross, Select Fragments of Aristotle, p. 27). That is to say, ‘If we ought to philosophise, then we ought to philosophise; and if we ought not to philosophise, then we ought to philosophise (i.e. in order to justify this view); in any case, therefore, we ought to philosophise’. So far as I know, this is the first appearance in philosophical literature of a pattern of argument that became popular among the Jesuits of the seventeenth century under the name of the consequentia mirabilis and inspired Saccheri's work Euclides ab Omni Naevo Vindicatus, in which theorems of non-Euclidean geometry were proved for the first time. The later history has been told by G. Vailati (in his article on Saccheri's Logica Demonstrativa, ‘Di un’ opera dimenticata del P. Gerolamo Saccheri’, reprinted in his Scritti, 1911, pp. 477–84), G. B. Halsted (in the preface to his 1920 edition of Saccheri's Euclides), and J. -Łukasiewicz (in his ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls’, Comptes Rendus des séances de la société des sciences et des lettres de Varsovie, Classe III, Vol. xxiii, 1930, p. 67). In this note I wish to consider only the early history of the argument and in particular a curious criticism of it which appears in Aristotle's Prior Analytics.

Isn’t all loss consequential?

2015 ◽  
Vol 7 (3) ◽  
pp. 176-194 ◽  
Author(s):  
Adam Connell ◽  
Jim Mason
Keyword(s):  
Sources Of Information ◽  
Breach Of Contract ◽  
Content Type ◽  
Legal Principles ◽  
Secondary Sources ◽  
Legal Position ◽  
Construction Law ◽  
Definition Of ◽  
First Time

Purpose – The purpose of this paper is to demystify the meaning of the term “consequential loss” in relation to the practice of construction law. Parties may have different understandings of the term and typically an exclusion clause will not solely relate to consequential loss, but will also include other heads of losses for which the party will not be liable for, such as loss of profit, loss of revenue and loss of business. Design/methodology/approach – The question emerges as to whether the term consequential loss has a definitive legal meaning in its own right. This study seeks to ascertain the definition of the term consequential loss within the construction industry through a review of the legal position regarding liability for breach of contract and consequential loss through the consideration of the case law relating to this topic and the associated secondary sources of information. Findings – The study concludes by elucidating a clear interpretation of the term consequential loss and guidance of how it should be used in contract law. Originality/value – Recent cases and established authorities are considered together for the first time in this work which assists in the development of legal principles of direct and indirect losses and the determination of how they apply to the built environment.

Sign in / Sign up

Export Citation Format

Share Document