SYNTHETIC REPRESENTATION OF THE

Vyacheslav Rustamyan; E. V. Bayanov; R. Slavin

doi:10.12737/2308-4898-2023-11-3-12-18

UDC 514.18

SYNTHETIC REPRESENTATION OF THE "OBLIQUE SYMMETRY" TRANSFORMATION USING THE EXAMPLE OF AN ELLIPSE

Published in Geometry & Graphics · Volume 11, Issue 3 · Pages 12–18 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/2308-4898-2023-11-3-12-18

Received: 25.12.2023 Accepted: 25.12.2023 Published: 25.12.2023 Language of publication: ru-RU

Authors

Vyacheslav Rustamyan

E. V. Bayanov

R. Slavin

Vyacheslav Rustamyan

MIREA — Moscow Technological University Senior Lecturer · Russian Federation · Москва

E. V. Bayanov

Academic degree: candidate of physical and mathematical sciences

Novosibirsk State Technical University Head · Novosibirsk

R. Slavin

Academic degree: candidate of technical sciences

Astrakhan State Technical University Associate Professor · Astrakhan

ABSTRACT

Geometric transformations play a pivotal role in computer graphics, determining the position and shape of objects. In machine learning, they are applied for processing and analyzing data, such as in images. In geometric surface modeling, they are utilized for the creation and transformation of three-dimensional forms. In physics, geometric transformations assist in describing the motion of objects in space and time. The aim of this work is to analyse and study the geometric transformation known as "oblique symmetry." Primarily, the article seeks to elucidate a number of important properties of this transformation, expanding the field of knowledge in perspective-affine correspondence. Throughout the study, the principal directions of oblique symmetry are identified, and their relationship with the axis and direction of the transformation is established. It is crucial to emphasise that the analysis makes it evident that the axis and the direction of symmetry are equivalent and interchangeable. Additionally, the article addresses the challenge of transforming an arbitrary ellipse, defined by its semi-axes, into a circle of equal area. In this context, a method is proposed to determine the axis and direction of oblique symmetry for a given ellipse. Based on the results obtained and the analysis conducted, the authors propose a geometric algorithm that provides the capacity to resolve positional problems in the field of descriptive geometry. This algorithm also offers a novel method for constructing ellipses with given semi-axes, which holds practical significance in various engineering and geometric issues. In the conclusion of the article, a specific example of applying the developed method is provided, clearly demonstrating its practical value and real capabilities in solving positional problems in the field of descriptive geometry. Moreover, directions for future research in the field of shape formation are suggested, utilising the "oblique symmetry" transformation in the spaces and .

KEYWORDS

affine correspondence, oblique symmetry transformation, positional problems, geometric transformations, conics, descriptive geometry

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Rustamyan V. V., Bayanov E. V., Slavin R. B. SYNTHETIC REPRESENTATION OF THE "OBLIQUE SYMMETRY" TRANSFORMATION USING THE EXAMPLE OF AN ELLIPSE // Geometry & Graphics. 2023. no. 3. pp. 12-18. DOI: https://doi.org/10.12737/2308-4898-2023-11-3-12-18. (Date of access 13.11.2026).

MLA

Vyacheslav Rustamyan, Bayanov E. V., Slavin R. B.. "SYNTHETIC REPRESENTATION OF THE "OBLIQUE SYMMETRY" TRANSFORMATION USING THE EXAMPLE OF AN ELLIPSE." Geometry & Graphics 11.3 (2023): 12-18. DOI: https://doi.org/10.12737/2308-4898-2023-11-3-12-18.

APA

Rustamyan V. V., Bayanov E. V., Slavin R. B. (2023). SYNTHETIC REPRESENTATION OF THE "OBLIQUE SYMMETRY" TRANSFORMATION USING THE EXAMPLE OF AN ELLIPSE. Geometry & Graphics, 11(3), 12-18. https://doi.org/10.12737/2308-4898-2023-11-3-12-18

Chicago

Vyacheslav Rustamyan, Bayanov E. V., Slavin R. B.. "SYNTHETIC REPRESENTATION OF THE "OBLIQUE SYMMETRY" TRANSFORMATION USING THE EXAMPLE OF AN ELLIPSE." Geometry & Graphics 11, no. 3 (2023): 12-18. https://doi.org/10.12737/2308-4898-2023-11-3-12-18.

WIKI

{{cite journal | author = Rustamyan V. V., Bayanov E. V., Slavin R. B. | title = SYNTHETIC REPRESENTATION OF THE "OBLIQUE SYMMETRY" TRANSFORMATION USING THE EXAMPLE OF AN ELLIPSE | journal = Geometry & Graphics | year = 2023 | volume = 11 | issue = 3 | pages = 12-18 | doi = 10.12737/2308-4898-2023-11-3-12-18 }}

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