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[7] viXra:2010.0228 [pdf] submitted on 2020-10-28 21:39:06
Authors: Hiroshi Okumura, Saburou Saitoh
Comments: 14 Pages.
In this paper, we will discuss Euclidean geometry from the viewpoint of the division by zero calculus with typical examples. Where is the point at infinity? It seems that the point is vague in Euclidean geometry in a sense. Certainly we can see the point at infinity with the classical Riemann sphere. However, by the division by zero and division by zero calculus, we found that the Riemann sphere is not suitable, but D\"aumler's horn torus model is suitable that shows the coincidence of the zero point and the point at infinity. Therefore, Euclidean geometry is extended globally to the point at infinity. This will give a great revolution of Euclidean geometry. The impacts are wide and therefore, we will show their essence with several typical examples.
Category: Geometry
[6] viXra:2010.0216 [pdf] submitted on 2020-10-27 10:43:42
Authors: Antoine Balan
Comments: 2 Pages.
We define a cohomology with values in an algebra fiber bundle.
Category: Geometry
[5] viXra:2010.0132 [pdf] replaced on 2021-05-25 14:00:21
Authors: Antonio Leon
Comments: 50 Pages.
This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.
Category: Geometry
[4] viXra:2010.0128 [pdf] replaced on 2024-04-26 13:10:14
Authors: Dmitriy Skripachov
Comments: 6 Pages.
A new approach in non-Euclidean geometry is obtained by adding types of deformation of the coordinate system. The concept of curvature is clarified taking into account the type of deformation. Such deformations can be stretching-compression, bending, shear, and torsion. The straightness of reference systems, traditional for non-Euclidean geometry, remains unchanged.
Category: Geometry
[3] viXra:2010.0110 [pdf] submitted on 2020-10-16 10:40:05
Authors: Franz Hermann
Comments: 9 Pages.
It is known that it is impossible to leave the hyperbolic geometry induced by the absolute as a second-order curve on the projective real plane. However, using the properties and methods of some imaginary hyperbolic geometry, it is possible to "teleport" a straight line segment located on a plane in hyperbolic geometry to another part of the same plane where the elliptic geometry is induced. Read about this in our article.
Category: Geometry
[2] viXra:2010.0047 [pdf] submitted on 2020-10-08 08:41:28
Authors: Franz Hermann
Comments: 15 Pages.
Данная работа являет собой новое направление в обширном геометрическом разделе, который носит название «Геометрические преобразования». Представление геометрического преобразования в виде векторной функции позволяет рассматривать некоторые вопросы, которые ранее здесь просто не могли бы даже возникнуть. Мы имеем ввиду прежде всего алгебру некоторых геометрических преобразований и формулы их композиций, которые позволяют построить новый математический аппарат в геометрии преобразований.
This work is a new direction in the extensive geometric section, which is called "Geometric transformations". The representation of a geometric transformation as a vector function allows us to consider some questions that previously could not even arise here. We mean first of all the algebra of certain geometric transformations and formulas of their compositions, which allow us to build a new mathematical apparatus in the geometry of transformations.
Category: Geometry
[1] viXra:2010.0028 [pdf] submitted on 2020-10-05 11:44:07
Authors: Franz Hermann
Comments: 33 Pages.
Сегодня практически в любом учебнике по аналитической геометрии есть раздел, посвящённый коническим сечениям. Великий математик древности Евклид когда-то написал сочинение «Начала конических сечений» (к сожалению до нас не дошедшее). Другой великий математик древности Аполлоний Пергский главный труд своей жизни так и озаглавил «Конические сечения». До наших дней сохранились семь из восьми книг этого сочинения. Вопрос конических сечений – кривых второго порядка – с древних времён интересовал человечество. И вопрос этот окончательно не закрыт до сих пор (например, некоторые особенности задачи Аполлония). В настоящей работе мы познакомим читателя с ещё одним взглядом на вопрос конических сечений, который мы назвали «Введение в теорию касательных сфер».
Today, almost every textbook on analytic geometry has a section on conical sections. The great mathematician of antiquity Euclid once wrote the essay "Beginnings of Conical Sections" (unfortunately not extant). Another great mathematician of antiquity, Apollonius of Perga, titled the main work of his life "Conical sections". Seven of the eight books of this work have survived to this day. The question of conical sections - curves of the second order - has been of interest to mankind since ancient times. And this question has not yet been completely closed (for example, some features of the Apollonius problem). In this paper, we will acquaint the reader with another look at the issue of conic sections, which we called "Introduction to the theory of tangent spheres".
Category: Geometry
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