Properties of Dupin Cyclide and Their Application. Part 4: Applications

N. Sal'kov

doi:10.12737/18055

PROPERTIES OF DUPIN CYCLIDE AND THEIR APPLICATION. PART 4: APPLICATIONS Properties of Dupin Cyclide and Their Application. Part 4: Applications

АА Аа

Published in Geometry & Graphics · Volume 4, Issue 1 · Pages 21–33 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/18055

Received: 02.03.2016 Accepted: 17.03.2016 Published: 17.03.2016 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 and second parts of the work there were considered mainly properties of Dupin cyclide, and given some examples of their application: three ways of solving the problem of Apollonius using only compass and ruler, using the identified properties of cyclide; it is determined that the focal surfaces of Dupin cyclid are degenerated in the lines and represent curves of the second order – herefrom Dupin cyclide can be defined by conic curve and a sphere whose center lies on the focal curve. Polyconic compliance of these focal curves is identified. The formation of the surface of the fourth order on the basis of defocusing curves of the second order is shown. In this issue of the journal the reader is invited to consider the practical application of Dupin cyclide’s properties. The proposed solution of Fermat’s classical task about the touch of the four spheres by the fifth with a ruler and compass, i.e., in the classical way. This task is the basis for the problem of dense packing. In the following there is an application of Dupin cyclide as a transition pipe element, providing smooth coupling of pipes of different diameters in places of their connections. Then the author provides the examples of Dupin cyclide’s application in the architecture as a shell coating. It is shown how to produce membranes from the same cyclide’s modules, from different modules of the same cyclide, from the modules of different cyclides, from cyclides with the inclusion of other surfaces, special cases of cyclides in the educational process. The practical application of the last problem found the place in descriptive geometry at the final geometrical education of architects in the "Construction of surfaces". Here such special cased of cyclides as conical and cylindrical surfaces of revolution.

KEYWORDS

descriptive geometry, cyclic surfaces, canal surface, Dupin cyclide, Fermat’s task, shell, architecture.

Text

В работах [23–25] были рассмотрены основные геометрические свойства цикдиды Дюпена [4–6; 8; 9; 11; 12; 20; 21; 29; 30]. В первой части [23] предлагаемой работы в качестве практического приложения циклид Дюпена рассматривалось построение окружности, касательной к трем данным окружностям — всемирно известная классическая задача Аполлония [9], когда данные окружности имели действительные радиусы.

Во второй части работы [24] рассмотрение свойств циклид Дюпена было продолжено. Предложена и доказана возможность задания циклиды Дюпена произвольным эллипсом в качестве линии центров множества образующих сфер и сферой с центром, принадлежащим этому эллипсу. Доказана достаточность этих сведений для построения циклиды Дюпена. Геометрически доказано, что фокальные линии циклид представляют собой не что иное, как кривые второго порядка. Дано графоаналитическое представление фокальных линий циклид. Показано поликоническое соответствие фокальных линий циклид Дюпена, которое рассмотрено во всех четырех случаях. Предложено формирование гиперболической поверхности четвертого порядка с использованием одной или двух первичных кривых второго порядка, в данном случае эллипсов.

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JATS XML

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 Dupin Cyclide and Their Application. Part 4: Applications // Geometry & Graphics. 2016. no. 1. pp. 21-33. DOI: https://doi.org/10.12737/18055. (Date of access 07.10.2026).

MLA

N. Sal'kov. "Properties of Dupin Cyclide and Their Application. Part 4: Applications." Geometry & Graphics 4.1 (2016): 21-33. DOI: https://doi.org/10.12737/18055.

APA

Sal'kov N. A. (2016). Properties of Dupin Cyclide and Their Application. Part 4: Applications. Geometry & Graphics, 4(1), 21-33. https://doi.org/10.12737/18055

Chicago

N. Sal'kov. "Properties of Dupin Cyclide and Their Application. Part 4: Applications." Geometry & Graphics 4, no. 1 (2016): 21-33. https://doi.org/10.12737/18055.

WIKI

{{cite journal | author = Sal'kov N. A. | title = Properties of Dupin Cyclide and Their Application. Part 4: Applications | journal = Geometry & Graphics | year = 2016 | volume = 4 | issue = 1 | pages = 21-33 | doi = 10.12737/18055 }}

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PROPERTIES OF DUPIN CYCLIDE AND THEIR APPLICATION. PART 4: APPLICATIONS Properties of Dupin Cyclide and Their Application. Part 4: Applications

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