Properties of Cyclide Dyupen and Their Application. Part 2

N. Sal&#039;kov

doi:10.12737/12164

PROPERTIES OF CYCLIDE DYUPEN AND THEIR APPLICATION. PART 2 Properties of Cyclide Dyupen and Their Application. Part 2

АА Аа

Published in Geometry & Graphics · Volume 3, Issue 2 · Pages 9–22 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/12164

Received: 10.06.2015 Accepted: 10.06.2015 Published: 10.06.2015 Language of publication: RUS

Authors

N. Sal'kov

Academic degree: candidate of technical sciences

Science rang: professor

MIREA – Russian technological university Russian Federation

ABSTRACT

In the first part of this paper has discussed about the basic properties of cyclide Dupin, and has gave some examples of their applications: three ways of solving the problem of Apollonius exclusively by means of compass and ruler, using identified properties cyclide Dupin, that is, given a classical solution of the problem. In the second of part of the work continued consideration of the properties of cyclide Dupin. Proposed and proved the possibility ask cyclide Dupin arbitrary ellipse as the center line of the forming a plurality of spheres and a sphere with the center belong - ing to the ellipse. Proved the adequacy of this information is used to build the cyclide Dupin. Geometrically proved that the focal line of cychlid are not that other, as curves of the second order. Given the graphical representation of the focal lines of cychlid. Shown polyconic compliance focal lines of cichlid of Dupin, which is considered in all four cases. The proposed formation of the hyperbolic surfaces of the fourth order with one or two primary curves of the second order, in this case ellipses. Apply sofocus this ellipse the hyperbola. Although the primary focus of the ellipse lying in the plane of the hoe, with the center coinciding with the origin of coordinates, is stationary, and the coordinate system rotates around the z axis. Then the points of intersection of the rotating coordinates x and y with a fixed ellipse specify new values for the major and minor axis of the ellipse with resultant changes in the form defocuses of the hyperbola. Although this modeling is not directly connected with Cychlidae Dupin, but clearly follows from the properties of its focal curves – curves of the second order. Withdrawn Equations of the surface and its throat.

KEYWORDS

descriptive geometry, circular surface, Kanaloa surface, a Dupin cyclide, the problem of Apollonius, the task Farm.

Text

В первой части предлагаемой работы [20] рассматривался вопрос об основных свойствах циклиды Дюпена [5; 6; 8; 10–12; 23], а также приводились некоторые примеры их применения: три способа решения задачи Аполлония [9] исключительно при помощи циркуля и линейки, используя выявленные свойства циклиды, т.е. показывалось классическое решение задачи. Во второй части работы продолжено рассмотрение свойств циклиды Дюпена. При исследовании были применены методы аналитической [3], проективной [21] и элементарной [1; 2] геометрий. Методам аналитической геометрии в статьях придается особое значении [5; 6; 8; 10–18; 23], поскольку они позволяют получить математически точное решение.

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Zamestitel' glavnogo redaktora, chlen redsoveta

Vladimir Vyshnyepolskiy

Glavnyy redaktor

A. A. Ishchenko

Vypuskayuschiy redaktor

Dar'ya Sklyankina

501@infra-m.ru

Deputy editor-in-chief, member of the editorial Board

N. Sal'kov

nikolaysalkov@mail.ru

GOST

Sal'kov N. A. Properties of Cyclide Dyupen and Their Application. Part 2 // Geometry & Graphics. 2015. no. 2. pp. 9-22. DOI: https://doi.org/10.12737/12164. (Date of access 25.09.2026).

MLA

N. Sal'kov. "Properties of Cyclide Dyupen and Their Application. Part 2." Geometry & Graphics 3.2 (2015): 9-22. DOI: https://doi.org/10.12737/12164.

APA

Sal'kov N. A. (2015). Properties of Cyclide Dyupen and Their Application. Part 2. Geometry & Graphics, 3(2), 9-22. https://doi.org/10.12737/12164

Chicago

N. Sal'kov. "Properties of Cyclide Dyupen and Their Application. Part 2." Geometry & Graphics 3, no. 2 (2015): 9-22. https://doi.org/10.12737/12164.

WIKI

{{cite journal | author = Sal'kov N. A. | title = Properties of Cyclide Dyupen and Their Application. Part 2 | journal = Geometry & Graphics | year = 2015 | volume = 3 | issue = 2 | pages = 9-22 | doi = 10.12737/12164 }}

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PROPERTIES OF CYCLIDE DYUPEN AND THEIR APPLICATION. PART 2 Properties of Cyclide Dyupen and Their Application. Part 2

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