THE CONNECTION BETWEEN THE PARABOLA AND THE ELLIPSE ON THE SURFACE OF THE SPHERE
Abstract
The sphere and the plane have common features. In both, the Gaussian curvature is constant: in a plane it is equal to zero, and in a sphere it depends on the radius. As a result, spherical curves can slide on the surface of a sphere in the same way that flat curves can slide in a plane. Some properties of flat curves are also characteristic of their spherical counterparts. If the profile of a tooth of a cylindrical gear is outlined along the involute of a circle, then the tooth of a conical gear is also outlined along a curve, which is a spherical analogue of the involute of a circle. In addition, two ellipses can roll over each other without slipping if their axes of rotation are located at the foci. The same applies to spherical ellipses, only unlike flat ellipses, in which the axes of rotation are parallel, in spherical ellipses they intersect at the center of the sphere. This similarity between flat curves and their spherical counterparts is used to design spherical mechanisms. The article considers the construction of a curve – a spherical analog of a parabola. The basis is the definition of a parabola as a geometric location of points equidistant from a fixed point – the focus of the parabola and from a straight line – the directrix. The equator is taken as the directrix on the sphere, as an analogue of a straight line on a plane. For the convenience of analytical calculations, a sphere of unit radius is taken. In this case, the lengths of the arcs are measured by angles. According to the derived equations, spherical parabolas were constructed, which, unlike flat ones, are closed. For a parabola on a plane, all rays coming from the focus are reflected from the parabola and form a bundle of parallel lines. The same thing happens on a spherical parabola, with the difference that the analog of parallel lines is a set of meridians that intersect at the pole. According to this property, a spherical parabola is similar to a spherical ellipse, in which rays emanating from one pole, after reflection, enter the other pole. The major axis of the ellipse in the angular dimension can take values up to 180°. It is mathematically proven that in the case when the major axis of the ellipse is equal to 90°, then the spherical ellipse is simultaneously a spherical parabola. Thus, a spherical parabola is a partial case of a spherical ellipse. The internal equation of a spherical parabola in curvilinear coordinates and its parametric equations have been compiled. Based on the obtained equations, parabolas with different values of the focal parameter were constructed. The condition under which a spherical parabola turns into a circle has been found.
References
2. Kresan, T., Ahmed, A., Pylypaka, S., Volina, T., Semirnenko, S., Trokhaniak, V., & Zakharova, I. (2023). Construction of spherical non-circular wheels formed by symmetrical arcs of loxodrome. Eastern-European Journal of Enterprise Technologies, 2023, 1 (1–121), 44–50.
3. Kresan, T., Pylypaka, S., Ruzhylo, Z., Rogovskii, I., & Trokhaniak, O. (2022). Construction of conical axoids on the basis of congruent spherical ellipses. Archives of Materials Science and Engineeringthis link is disabled, 2022, 113 (1), 13–18.
4. Nesvidomin, A., Pylypaka, S., Volina, T., Kalenyk, M., Shuliak, I., Semirnenko, Y., Tarelnyk, N., Hryshchenko, I., & Kholodniak, Y. (2023). Constructing geometrical models of spherical analogs of the involute of a circle and cycloid. Eastern-European Journal of Enterprise Technologies, 4 (7 (124)), 6–12.
5. Wang, Yo., Chang, Yu. (2020). Mannheim curves and spherical curves. International Journal of Geometric Methods in Modern Physics, 17 (07), 2050101.
6. Yayli, Yu., Saracoglu, S. (2011). Some Notes on Dual Spherical Curves. Journal of Informatics and Mathematical Sciences, 3 (2), 177–189.
7. Volina, T. M., Pylypaka, S. F., & Nesvidomin, A. V. (2021). Rukh chastynky po sferychnomu sehmentu z vertykalnymy radialno vstanovlenymy lopatkamy [Movement of a particle along a spherical segment with vertical radially installed blades]. Mechanics and mathematical methods, IІІ, 1, 27–36 (in Ukrainian).
8. Danylevskyi, M. P., Kolosov, A. I., & Yakunin, A. V. (2011). Osnovy sferychnoi heometrii ta tryhonometrii: navch. posib. [Basics of spherical geometry and trigonometry: a textbook]. Khark. nats. akad. misk. hosp-va, Kh.: KhNAMH, 92 p. (in Ukrainian).
9. Pylypaka, S. F., Hryshchenko, I. Yu., & Kresan, T. A. (2018). Modeliuvannia smuh rozghortnykh poverkhon, dotychnykh do poverkhni kuli [Modeling of striping surfaces tangent to the surface of the sphere]. Prykladni pytannia matematychnoho modeliuvannia, Kherson: OLDI-PLIUS, 1, 81–88 (in Ukrainian).
10. Pylypaka, S. F., Zakharova, T. M. (2014). Konstruiuvannia sferychnykh kryvykh u funktsii naturalnoho parametra [Construction of spherical curves as a function of a natural parameter]. Naukovyi visnyk Melitopolskoho derzhavnoho pedahohichnoho universytetu imeni Bohdana Khmelnytskoho. Seriia: Matematyka. Heometriia. Informatyka. Melitopol: Vydavnytstvo MDPU im. B. Khmelnytskoho, 1, 137–145 (in Ukrainian).
11. Pylypaka, S. F., Nesvidomin, A. V. (2023). Formoutvorennia sferychnykh epitsykloid pry obkochuvanni rukhomoho konusa po nerukhomomu [Formation of spherical epicycloids when rolling a moving cone on a stationary one]. Visnyk Khersonskoho natsionalnoho tekhnichnoho universytetu, 2 (85), 65–70 (in Ukrainian).
12. Pylypaka, S. F. (2008). Analitychne konstruiuvannia prostorovykh ta sferychnykh kryvykh u funktsii vlasnoi duhy [Analytical construction of spatial and spherical curves as a function of their own arc]. Heometrychne ta kompiuterne modeliuvannia, 21. Kharkiv: KhDUKhT, 100–105 (in Ukrainian).
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