Constructing a method for the geometrical modeling of the lame superellipses in the oblique coordinate systems

Authors

DOI:

https://doi.org/10.15587/1729-4061.2020.201760

Keywords:

Lamé superellipse, geometrical modeling, oblique coordinate system, angle of inclination of the tangent, curvature distribution

Abstract

The elliptic curves possess a certain disadvantage related to that at the point of intersection with the coordinate axes the ellipses have tangents perpendicular to these axes. However, such a situation is undesirable for some practical applications of ellipses. It can be prevented by modeling the specified curves in oblique coordinates, which, in turn, are related to a certain original Cartesian coordinate system. The Lamé superellipses are understood to be the curves whose equations include the exponents that differ from those inherent in regular ellipses. Variating these exponents can produce a wide range of different curves. This paper has proposed a method for the geometric modeling of superellipses in the oblique coordinate systems. The source data for modeling are the coordinates of the two points with the known angles of the tangent slope. The accepted axes of the oblique coordinate system are the straight lines drawn as follows. Through the first point, a line parallel to the tangent at the second point is built, and at the second point, a line parallel to the tangent at the first point is constructed. It has been shown that these operations could yield the desired values of tangent angles at intersection points of the superellipse with axial lines. It has been proven that the superellipse arc could be drawn through a third given point with the required angle of the tangent; that, however, would require determining the exponents in the superellipse equation by a numerical method. Such a situation occurs, for example, when designing the projected profiles of axial turbine blades. Based on the proposed method of modeling the superellipse curves, a computer code has been developed that could be used in describing the contours of components applied in the technologically complex industries

Author Biographies

Valeriy Borisenko, Mykolaiv V. O. Sukhomlynskyi National University Nikolska str., 24, Mykolaiv, Ukraine, 54030

Doctor of Technical Sciences, Professor

Department of Information Technology

Serhiy Ustenko, Odessa National Polytechnic University Shevchenka ave., 1, Odessa, Ukraine, 65044

Doctor of Technical Sciences, Associate Professor

Department of IT Designing Training

Iryna Ustenko, Admiral Makarov National University of Shipbuilding Heroiv Ukrainy ave., 9, Mykolaiv, Ukraine, 54025

PhD, Associate Professor

Department of Automated Systems Software

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Published

2020-04-30

How to Cite

Borisenko, V., Ustenko, S., & Ustenko, I. (2020). Constructing a method for the geometrical modeling of the lame superellipses in the oblique coordinate systems. Eastern-European Journal of Enterprise Technologies, 2(4 (104), 51–59. https://doi.org/10.15587/1729-4061.2020.201760

Issue

Vol. 2 No. 4 (104) (2020): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2020 Valeriy Borisenko, Serhiy Ustenko, Iryna Ustenko

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