Graphic-Analytical Researches of Involution

O. Grafskiy; A. V. Usmanov; Aleksandr A. Holodilov

doi:10.12737/25118

UDC

GRAPHIC-ANALYTICAL RESEARCHES OF INVOLUTION Graphic-Analytical Researches of Involution

АА Аа

Published in Geometry & Graphics · Pages 3–11 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/25118

Received: 05.05.1905 Accepted: 05.05.1905 Published: 05.05.1905 Language of publication: RUS

Authors

O. Grafskiy

A. V. Usmanov

Aleksandr A. Holodilov

O. Grafskiy

Academic degree: doctor of technical sciences

Far Eastern State Transport University Professor · Russian Federation

A. V. Usmanov

Aleksandr A. Holodilov

Far Eastern State Transport University Lecturer · “Computation Technique” Department · Russian Federation · Khabarovsk

ABSTRACT

Known projective transformations, namely their private types such as harmonism and involution are considered. It is known that projective transformations are collinear, at their performance the order, the cross ratio of fours of elements (on a straight line — the cross ratio of four points, in a bunch of straight lines — the cross ratio of this bunch’s four straight lines, this property (invariant) is similarly preserved for a bunch of planes, i.e. in considering of first step forms) is preserved. At a constructive approach to such transformations there are some ways for definition of position for corresponding elements which students use when studying discipline "Affine and projective geometry" on preparation profiles 09.03.01 — “CAD Systems" and 09.03.03 — “Applied Informatics in Design”. The received constructions are checked by analytical calculations, proceeding from known dependences for harmonism and involutions. In such a case results both for a range of points, and for a bunch of straight lines which pass through these points are analytically compared. The provided computational and graphic work contains three sections: prospects, harmonism and involution, and is carried out by students on individual options with application of the graphic editor Microsoft Visio or the graphic package CO MPAS voluntary. In the present paper some constructions in definition of corresponding points in elliptic and hyperbolic involution are considered, some of these constructions are published for the first time. Besides, a proposition has been formulated: in a rectangular coordinate system the work for coordinates of two points related to a circle intersection with one coordinate axis is equal to the product for coordinates of two other points related to this circle intersection with the other coordinate axis. This proposition is fairly for imaginary points of circle intersection with coordinate axes as well.

KEYWORDS

transformations; harmonism; hyperbolic and elliptic involution; cross ratio of four points; quadratic map.

Text

Известно, что инволюция (рассматриваются, например, два совмещенных проективных ряда соответственных точек на прямой линии — инволюционный ряд точек) бывает трех типов [2; 3; 9–11; 21]:
• гиперболическая, когда две двойные точки являются действительными;
• эллиптическая, здесь две двойные точки являются мнимыми;
• параболическая, в этом случае имеем две совпавшие двойные точки.

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GOST

Grafskiy O. A., Usmanov A. V., Holodilov A. A. Graphic-Analytical Researches of Involution // Geometry & Graphics. 2017. no. 1. pp. 3-11. DOI: https://doi.org/10.12737/25118. (Date of access 11.09.2026).

MLA

O. Grafskiy, Usmanov A. V., Holodilov A. A.. "Graphic-Analytical Researches of Involution." Geometry & Graphics 5.1 (2017): 3-11. DOI: https://doi.org/10.12737/25118.

APA

Grafskiy O. A., Usmanov A. V., Holodilov A. A. (2017). Graphic-Analytical Researches of Involution. Geometry & Graphics, 5(1), 3-11. https://doi.org/10.12737/25118

Chicago

O. Grafskiy, Usmanov A. V., Holodilov A. A.. "Graphic-Analytical Researches of Involution." Geometry & Graphics 5, no. 1 (2017): 3-11. https://doi.org/10.12737/25118.

WIKI

{{cite journal | author = Grafskiy O. A., Usmanov A. V., Holodilov A. A. | title = Graphic-Analytical Researches of Involution | journal = Geometry & Graphics | year = 2017 | volume = 5 | issue = 1 | pages = 3-11 | doi = 10.12737/25118 }}

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