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Constructing a Method for the Geometrical Modeling

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Constructing a Method for the Geometrical Modeling Mathematics and cybernetics – applied aspects Еліптичним кривим притаманний певний не- UDC 514.18 долік, пов’язаний з тим, що в точках перетину DOI: 10.15587/1729-4061.2020.201760 з осями координат еліпси мають дотичні пер- пендикулярні до цих осей. Проте в деяких прак- тичних застосуваннях еліпсів подібна ситуація є небажаною. Запобігти цьому можна моделю- CONSTRUCTING ванням вказаних кривих у косокутних координа- тах, які, в свою чергу, віднесені до деякої вихідної A METHOD FOR ортогональної координатної системи. Під супе- реліпсами Ламе розуміються криві, в рівняннях THE GEOMETRICAL яких застосовуються показники степенів, від- мінні від двох, що є притаманним для звичайних MODELING OF THE LAME еліпсів. Варіюванням цими показниками степе- нів можна отримати широке коло різноманітних SUPERELLIPSES IN THE кривих. У цій роботі запропоновано метод геоме- тричного моделювання супереліпсів у косокутних координатних системах. Вихідними даними для OBLIQUE COORDINATE моделювання є координати двох точок з відомими в них кутами нахилу дотичних. За вісі косокутної SYSTEMS системи координат приймаються прямі, прове- дені наступним чином. Через першу точку буду- ється пряма паралельно дотичній в другій точці, V. Borisenko а в другій точці – пряма паралельно дотичній Doctor of Technical Sciences, Professor в першій точці. Показано, що завдяки цим захо- Department of Information Technology дам можна забезпечити потрібні значення кутів Mykolaiv V. O. Sukhomlynskyi National University нахилу дотичних в точках перетину супереліпса Nikolska str., 24, Mykolaiv, Ukraine, 54030 з осьовими лініями. Доведено, що дугу суперелі- Е-mail: [email protected] пса можна проводити через третю задану точку S. Ustenko з потрібним в ній кутом нахилу дотичної, але це Doctor of Technical Sciences, Associate Professor потребує визначення числовим методом показни- Department of IT Designing Training ків степенів у рівнянні супереліпса. Подібна ситуа- ція має місце, наприклад, при розробці проектів Odessa National Polytechnic University профілів лопаток осьових турбін. На підставі Shevchenka ave., 1, Odessa, Ukraine, 65044 запропонованого методу моделювання дуг супере- Е-mail: [email protected] ліпсів розроблено комп’ютерний код, який можна I. Ustenko застосовувати при описі контурів виробів техно- PhD, Associate Professor логічно складних галузей промисловості Department of Automated Systems Software Ключові слова: супереліпс Ламе, геометрич- Admiral Makarov National University of Shipbuilding не моделювання, косокутна система координат, Heroiv Ukrainy ave., 9, Mykolaiv, Ukraine, 54025 кут нахилу дотичної, розподіл кривини E-mail: [email protected] Received date 20.02.2020 Copyright © 2020, V. Borisenko, S. Ustenko, I. Ustenko Accepted date 20.04.2020 This is an open access article under the CC BY license Published date 30.04.2020 (http://creativecommons.org/licenses/by/4.0) 1. Introduction London). It is known from astronomy that the planets in the solar system revolve around the Sun in orbits in the form of Elliptic curves are the generally known curves of the ellipses. In our time, thousands of artificial satellites move second order, which are characterized by the axial symme- around the Earth through elliptic orbits. try relative to the Ox and Oy axes, as well as the central The elliptical curves are understood to be the closed flat symmetry relative to the coordinate origin [1, 2]. The tech- lines, which can be obtained as the cross-section of a cylin- niques for constructing these curves using graphical methods der or a rotating cone by the plane inclined to their axis at are considered in descriptive geometry [3], engineering [4], a certain angle. It can also be the mapping of a circle onto and computer graphics [5]. The examples of the practical a plane not parallel to the plane of the circle location. A circle application of the elliptic curves in shipbuilding are given is a separate case of an ellipse. With the affine transformation in re ference book [6], in the theory of mechanisms and ma- of a circle, you can get an elliptical curve or just an ellipse. chines – in [7], in the construction of profiles of the axial The elliptical curves possess some specific benefits due, turbine blades – in [8]. Given their reflecting capability, the for example, to the monotony of a curvature change, an angle ellipses are widely used in architecture and building, parti- of inclination of the tangent, derivatives, etc. However, there cularly when erecting domes of palaces and cathedrals, as is a certain shortcoming in the ellipses that is related to the well as amphitheaters (for example, the «Hall of Secrets» angles of inclination of the tangents at the intersection points of the Alhambra in Granada and St. Peter’s Cathedral in of the curve with the coordinate axes. These angles accept 51 Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 2/4 ( 104 ) 2020 either zero values or are equal to 90°. However, for ellipses Equation (2) defines a closed curve, limited by a rectan- that are built in Cartesian coordinates, there is no way to gle with sides – a ≤ x ≤ a and – b ≤ y ≤ b. It is the generalized ensure arbitrary values of the angles of inclination of the equation of the ellipse, which, depending on the exponent tangent at intersections with the coordinate axes. value and the magnitudes of the semi-axes, makes it possible One possible way to ensure the arbitrary values of the to obtain a circle, an ellipse, a square, and a rectangle. At n = 1, angles of inclination of the tangents at the intersection points the curve degenerates into a straight line; at n = 2, we obtain of the curve with the coordinate axes is to model them in a regular ellipse, at n = 2/3 – the astroid (provided a = b). oblique coordinate systems, as well as to construct new ellip- Some of the features of the Lamé superellipse can be tic curves by transforming the already known curve. found in [10], in particular, the expressions for calculating These issues are of theoretical and practical significance. the arc length and the curve square. In the cited work, the su- They are relevant for those industries where the articles of perellipses are examined in the Cartesian coordinates not in complex geometric shapes are made (for example, in ship- the form of arcs, located in the region of positive coordinates, building, when describing waterlines, frames, battoks lines; in but in the so-called full form. the gas turbine engineering, when modeling air intakes, pro- Now it is difficult to determine who for the first time files of turbine blades). Thus, in the gas turbine industry, it offered to apply equation (2) with different exponent values, is important to ensure the estimated angles of a flow inlet to that is, to write it in a more general form: and out of the blade apparatus. At the same time, it is neces- mn sary to meet the conditions for a smooth transition between x y + = 1. (3) the angles of inclination of the tangents from the starting a b point to the endpoint of the modeled curve. It should be noted that recent years have seen significant Applying different exponent values provides even more qualitative changes in designing complex highly technologi- possibilities to constructing a set of various curves. cal products in various industries. There is a widespread shift Summing up, one can note that all the above equations from traditional graphic information processing to paperless are recorded in almost identical mathematical notation and technologies based on the digital descriptions of projected differ only in the values of the exponents. and manufactured objects. Computerized technologies make The most striking examples of the practical application of it possible to create numerical models of different objects. the Lam superellipse include the Aztec Stadium, built before A designer can view the physically non-existent object on the Olympic Games in Mexico City, and a square in Stockholm. the computer, get the desired geometric characteristics, make Paper [11] gives examples of constructing superellipses certain changes, prepare the production and, finally, produce with equal exponent values. Those examples demonstrate one or another product in modern machining centers. a change in the geometry of curves as the exponents gra- Geometric information about products must determine dually decrease. It is indicated that one of the squares in them in full, meet the requirements arising from the func- Stockholm has the shape of a superellipse. tional, structural, strength, ergonomic, aesthetic, operating, The examples of using superellipses in computer graphics technological, and other conditions. The most important are shown in work [12]. The specified curves are considered component of the information used in the manufacture of with equal exponents in the Cartesian coordinates. products in the technologically sophisticated, knowledge- Study [13] demonstrates that nature has many examples intensive industries is the geometric model of an object, which of plants whose transverse cross-section coincides in the contains a description of its shape, as well as the description of shape with the Lamé superellipses, which are termed the the connecting elements in the model. Lamé superellipses. The cited study considers superellipses with equal exponent values. Further results of the previous studies are given in [14], which considers the superellipse 2. Literature review and problem statement equations in the polar coordinate system and at different ex- ponent values. That advanced the circle of a variety of curves In the Cartesian coordinate system, the ellipse is de- whose examples found in the plant world. scribed by the following equation [9, 10]: Work [15] built on the search related to the description of the features in the plant world. It also uses the Lamé su- 22 x y perellipse with equal exponent values; that ellipse is formed + = 1, (1) a b in the Cartesian coordinates. Study [16] applies the Lamé superellipses to describe where a and b are the ellipse semi-axes whose equality trans- the symmetrical forms of bamboo leaves, which is crucial forms the ellipse into a circle.
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