Representations of Dupin Cyclides

N. Sal&#039;kov

doi:10.12737/article_59bfa354466be1.50763524

UDC 51

REPRESENTATIONS OF DUPIN CYCLIDES Representations of Dupin Cyclides

АА Аа

Published in Geometry & Graphics · Volume 5, Issue 3 · Pages 11–24 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/article_59bfa354466be1.50763524

Received: 18.09.2017 Accepted: 24.03.2024 Language of publication: RUS

Authors

N. Sal'kov

Academic degree: candidate of technical sciences

Science rang: professor

MIREA – Russian technological university Russian Federation

ABSTRACT

We know very little about such an interesting surface as Dupin cyclide. It belongs to channel surfaces, its special cases are tor, conical and cylindrical surfaces of rotation. It is known that Dupin cyclides are the only surfaces whose focal surfaces, that are surfaces consisting of sets of curvatures centers points, have been degenerated in second-order curves. Two sets give two confocal conics. That is why any study of Dupin cyclides is of great interest both scientific and applied. In the works devoted to Dupin cyclide and published in the "Geometry and Graphics" journal, are presented various properties of cyclides, and demonstrated application of these surfaces in various industries, mostly in construction. Based on the cyclides’ properties in 1980s have been developed numerous inventions relating to devices for drawing and having the opportunity to be applied in various geometric constructions with the use of computer technologies. In the present paper have been considered various options for representation of Dupin cyclides on a different basis – from the traditional way using the three given spheres unto the second-order curves. In such a case, if it is possible to represent four cyclides by three spheres, and when cyclide is represented by the second-order curve (konic) and the sphere their number is reduced to two, then in representation of cyclide by the conic and one of two cyclide’s axes a single Dupin cyclide is obtained. The conic itself without any additional parameters represents the single-parameter set of cyclides. Representations of Dupin cyclides by ellipse, hyperbola and parabola have been considered. The work has been sufficiently illustrated.

KEYWORDS

Dupin cyclide, surface constructing, second-order curves, descriptive geometry, computer graphics, engineering graphics.

Text

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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. Representations of Dupin Cyclides // Geometry & Graphics. 2017. no. 3. pp. 11-24. DOI: https://doi.org/10.12737/article_59bfa354466be1.50763524. (Date of access 24.09.2026).

MLA

N. Sal'kov. "Representations of Dupin Cyclides." Geometry & Graphics 5.3 (2017): 11-24. DOI: https://doi.org/10.12737/article_59bfa354466be1.50763524.

APA

Sal'kov N. A. (2017). Representations of Dupin Cyclides. Geometry & Graphics, 5(3), 11-24. https://doi.org/10.12737/article_59bfa354466be1.50763524

Chicago

N. Sal'kov. "Representations of Dupin Cyclides." Geometry & Graphics 5, no. 3 (2017): 11-24. https://doi.org/10.12737/article_59bfa354466be1.50763524.

WIKI

{{cite journal | author = Sal'kov N. A. | title = Representations of Dupin Cyclides | journal = Geometry & Graphics | year = 2017 | volume = 5 | issue = 3 | pages = 11-24 | doi = 10.12737/article_59bfa354466be1.50763524 }}

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REPRESENTATIONS OF DUPIN CYCLIDES Representations of Dupin Cyclides

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