Moskva, Moscow, Russian Federation
Moskva, Moscow, Russian Federation
Rotation is the motion of geometric objects along a circle. This is one of geometric techniques used to form lines and surfaces. In this paper has been considered the rotation of objects in a three-dimensional space around a straight axis. It is known that a straight line can be considered as a particular case of a circle with a radius equal to infinity. Such circle’s center is at infinite distance from the considered straight line segment. Then in the general case, the rotation axis is a closed curve, for example, a circle with a radius of finite magnitude. Rotation of a point around a straight axis now splits into two trajectories. One of them is a circle with a radius, the second is a straight line crossing with the axis, and the center of this trajectory is at an infinite distance from the point. The method of point rotation about an axis of finite radius was considered. Note that a circle is a special case of an ellipse. When the actual focus of the circle is stratified into two, the line itself loses its curvature constancy, and is called an ellipse. The point, rotating around the elliptical axis, is stratified into four ones, forming four circles (trajectories). Axis foci appearing in turn in the role of the main one determine two trajectories by each with a trivial and nontrivial center of rotation. We have considered the variant for arrangement of the generating circle so that its center coincided with one of the elliptic axis’s foci. The obtained surfaces are a pair of co-axial Dupin cyclides, since they have identical properties. Changing the circle generatrix radius, other things being equal, we get different types of closed cyclides.
rotation, form making, curvilinear axis, circle, ellipse, Dupin cyclide, cyclic surfaces, rotation axis, rotation trajectory.
Вращение — движение геометрических объектов по окружности. Это один из геометрических приемов, который используется для образования линий и поверхностей. Свойства поверхностей вращения рассмотрены в работах [2–4; 6; 11–17; 24].
1. Argunov B.I., Balk M.B. Geometricheskiye postroyeniya na ploskosti [Geometric constructions on the plane]. Moscow, Uchpedgiz Publ., 1957. 267 p.
2. Bermant A.F. Geometricheskiy spravochnik po matematike (Atlas krivykh) CH.1. [Geometric reference book on mathematics (Atlas of curves) Part. 1]. Moscow, ONGIZ NKTP Publ., 1937. 209 p.
3. Vinogradov V.N. Nachertatel'naya geometriya [Descriptive geometry]. Minsk, High School Publ., 1977. 367 p.
4. Gil'bert D., Kon-Fossen S. Nagljadnaya geometriya [Visual geometry]. Moscow, Leningrad, Obyedinennoe nauchno-tehnicheskoe izdatel'stvo NKTP SSSR, Glavnaya redakciya obshhetehnicheskoj literatury i nomografii Publ., 1936. 302 p.
5. Girsh A.G. Fokusy algebraicheskikh krivykh [Focuses of algebraic curves]. Geometriya i grafika [Geometry and Graphics]. V. 3, I. 3, pp. 4–17. (in Russian). DOI: 10.12737/14415.
6. Gordon V.O., Sementsov-Ogievsky M.A. Kurs nachertatel'noj geometrii [Course descriptive geometry]. Moscow: Higher School Publ., 1998. 272 p.
7. Grafskiy O.A., Doronina S.S., Galliulin N.Kh. Analiz postroyeniya krivykh vtorogo poryadka [Analysis of the con-struction of curves of the second order]. Nauchno-tekhnicheskoye i ekonomicheskoye sotrudnichestvo stran ATR v XXI veke: Materialy Vserossiyskoy nauchno-prakticheskoy konferentsii s mezhdunarodnym uchastiem, 22–24 aprelya 2009 g. [Scientific, technical and economic cooperation of the APR countries in the XXI century: Materials of the All-Russian Scientific and Practical Conference with International Participation, April 22–24, 2009]. Khabarovsk, FESTU Publ., 2009, pp. 165–168. (in Russian)
8. Grafskij O.A., Saenko O.V. Kasatel'naya k okruzhnosti [Tangent to a circle]. Nauchno-tehnicheskie problemy transporta, promyshlennosti i obrazovaniya: Trudy Vserossiyskoy nauchno-praktich. konferentsii [Scientific andtechnical problems of transport, industry and education:works of the All-Russian scientific and practical conference]. Khabarovsk, FESTU Publ., 2010, pp. 190–192. (in Russian)
9. Grafskij O.A., Saenko O.V. Obosnovanie postroeniya kasatelnoy k okruzhnosti i ellipsu [Justification of creation ofa tangent to a circle and an ellipse]. Nauchno-tehnicheskie problemy transporta, promyshlennosti i obrazovaniya: Trudy Vserossiyskoy nauchno-praktich. Konferentsii, 20–22 aprelya 2011 g. [Scientific andtechnical problems of transport, industry and education: works of the All-Russian scientific and practical conference, April 20–22, 2011]. Khabarovsk, FESTU Publ., 2011, pp. 14–18. (in Russian)
10. Gryaznov YA.A. Otsek kanalovoy poverkhnosti kak obraz tsilindra v rassloyayemom obrazovanii [A compartment of the channel surface as an image of a cylinder in a stratified formation]. Geometriya i grafika [Geometry and graphics]. 2013, V. 1, I. 3, pp. 17–19. DOI: 10.12737/6518. (in Russian)
11. Zelenin E.V. Nachertatel'naya geometriya i cherchenie [Descriptive geometry and drawing]. Moscow, Gosudarstvennoe izdatel'stvo tehniko-teoreticheskoj literatury Publ., 1953. (in Russian)
12. Ivanov G.S. Nachertatel'naya geometriya [Descriptive geometry]. Moscow, FGBOU VPO MGUL Publ., 2012. 340 p.
13. Ivanov G.S., Dmitrieva I.M. Teoreticheskie osnovy nachertatel'noy geometrii [On the tasks of descriptive geometry with imaginary solutions]. Geometriya i grafika [Geometry and Graphics]. 2015, I. 2, pp. 3–8. (in Russian). DOI: 10.12737/12163.
14. Klein F. Vysshaja geometriya [Higher geometry]. Moscow, Leningrad, GONTI Publ., 1939.
15. Monzh G. Nachertatel`naya geometriya [Descriptive geometry]. Moscow, Academy of Sciences of the USSR Publ., 1947. 291 p.
16. Posvyanskiy A.D. Kratkij kurs nachertatel'noj geometrii [A short course of descriptive geometry]. Moscow, Higher School Publ., 1970. 240 p.
17. Salkov N.A. Nachertatel'naya geometriya [Descriptive geometry]. Moscow, INFRA-M Publ., 2013. 184 p. (in Russian)
18. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast' 1 [Properties cyclid of Dupin and their application. Part 1]. Geometriya i grafika [Geometry and graphics]. 2015, V. 3, I. 1, pp. 16–25. DOI: 10.12737/10454. (in Russian)
19. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast' 2 [Properties cyclid of Dupin and their application. Part 2]. Geometriya i grafika [Geometry and graphics]. 2015, V. 3, I. 2, pp. 9–23. DOI: 10.12737/12164. (in Russian)
20. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast' 3 [Properties cyclid of Dupin and their application. Part 3]. Geometriya i grafika [Geometry and graphics]. 2015, V. 3, I. 4, pp. 3–14. DOI: 10.12737/17345. (in Russian)
21. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast' 4 [Properties cyclid of Dupin and their application. Part 4]. Geometriya i grafika [Geometry and graphics]. 2016, V. 4, I. 1, pp. 21–32. DOI: 10.12737/17347. (in Russian)
22. Sal'kov N.A. Tsiklida Dyupena i krivyye vtorogo poryadka. Chast' 1 [Cyclid Dupin and curves of the second order]. Geometriya i grafika [Geometry and graphics]. 2016, V. 4, I. 2, pp. 19–28. DOI: 10.12737/19829. (in Russian)
23. Sal'kov N.A. E`llips: kasatel'naya i normal` [Ellipse: tangent and normal]. Geometriya i grafika [Geometry and graphics]. 2013, V. 1, I. 1, pp. 35–37. DOI: 10.12737/2084. (in Russian)
24. Frolov S.A. Nachertatel`naya geometriya [Descriptive geometry]. Moscow, Mechanical Engineering Publ., 1978. 240 p.
25. Chetverukhin N.F. Proektivnaya geometriya [Projective geometry]. Moscow, Prosveshchenie Publ., 1969. 386 p.
