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

Valeriy Borisenko,Serhiy Ustenko,Iryna Ustenko

doi:10.15587/1729-4061.2020.201760

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 Lame 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

Highlights

Elliptic curves are the generally known curves of the second order, which are characterized by the axial symmetry relative to the Ox and Oy axes, as well as the central symmetry relative to the coordinate origin [1, 2]
The graphic data are the visual confirmation of the operability of the proposed method of geometric modeling of the Lamé superellipses in the oblique coordinates at two and three preset coordinates of the points and the angles of inclination of the tangents
The construction of the arc of a regular ellipse, which has the same exponent values equal to two, in the oblique coordinates makes it possible to obtain these arcs at the assigned angles of inclination of the tangents at the endpoints

Summary

Introduction

Elliptic curves are the generally known curves of the second order, which are characterized by the axial symmetry relative to the Ox and Oy axes, as well as the central symmetry relative to the coordinate origin [1, 2]. The examples of the practical application of the elliptic curves in shipbuilding are given in reference book [6], in the theory of mechanisms and machines – in [7], in the construction of profiles of the axial turbine blades – in [8]. Given their reflecting capability, the ellipses are widely used in architecture and building, parti cularly when erecting domes of palaces and cathedrals, as well as amphitheaters (for example, the «Hall of Secrets» of the Alhambra in Granada and St. Peter’s Cathedral in London). Thousands of artificial satellites move around the Earth through elliptic orbits

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