EPI-HYPOCYCLOIDS AND EPI-HYPOCYCLOIDAL CANAL SURFACES

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Abstract


In the article are regarded the curves - epiand hypocycloids, which are formed by the moving of the generating points, linked with the circles of the same radius and which are at the same time outside and inside of the unmoving circle. There is shown the relation of those curves. The moving of the circles with constant angle to the plane of the unmoving circle is also regarded. At full rotation of the moving circle the generating point linked with moving circle described a circle around the tangent of the unmoving circle. And the initial point laying in horizontal plane on epicycloid moving to the point on hypocycloid when the moving circle rotates on around the tangent of the unmoving circle. When the circle made a full rotation around the unmoving circle with full rotation around the tangent to the unmoving circle the epi-hypocycloidal cyclic surfaces are formed. In the article is proofed that the circles of the epi-hypocycloidal cyclic surfaces are the coordinate lines of the main curvatures of the surface and so the surfaces belongs to the class of canal surfaces. The drawings of the epi-hypocycloidal canal surfaces with different parameters - relation of the radius of the moving and unmoving circles λ, the position of the generating point μ - are shown.


About the authors

Vyacheslav N Ivanov

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: i.v.ivn@mail.ru
6 Miklukho-Maklaya St., Moscow, 117198, Russia

Doctor of Technical Sciences, Professor of the Department of Architecture and Civil Engineering, Engineering Academy, Peoples' Friendship University of Russia (RUDN University). Scientific interests: geometry, surface shaping and methods for calculating thin-walled structures of complex shapes

References

  1. Bronshtain I.N., Semendyaev K.A. (1962). Spravochnik po matematike [Reference book of higher mathematics]. For engineers and students of the VTUZes. Мoscow, GIFizMatlit Publ., 608. (In Russ.)
  2. Smirnov V.I. (1965). Kurs vysshei matematiki [Course of higher mathematics]. Мoscow, Nauka Publ., 1, 480. (In Russ.)
  3. Ivanov V.N., Romanova V.A. (2016). Konstruktsionnye formy prostranstvennykh konstruktsii. Vizualizatsiya poverkhnostei v sistemakh MathCad, AutoCad [Constructive forms of the space constructions. Visualization of the surfaces in the systems of MathCad and UutuCad]. Monograph. Moscow, FSV Publ., 412. (In Russ.)
  4. Lawrence J. Dennis. (1972). A catalog of special plane curves. Dover Publications, 161, 168–170, 175.
  5. Krivoshapko S.N., Ivanov V.N. (2015). Encyclopedia of Analytical Surfaces. Switzerland: Springer International Publishing, 752.
  6. Ivanov V.N., Mahmud H.S. (1990). Koordinatnaya set' linii krivizny epitro-khoidal'noi poverkhnosti [Coordinate system of the curvature lines of the epitrochoidal surface]. Investigation of structural mechanics of the space systems. Moscow, UDN Publ., 38–44. (In Russ.)
  7. Shulikovskiy V.I. (1963). Klassicheskaya differrentsial'naya geometriya [Classic differential geometry]. Мoscow, GIFML Publ., 540. (In Russ.)
  8. Ivanov V.N., Krivoshapko S.N. (2010). Analiticheskie metody rascheta obolochek nekanonicheskoi formy [Analitycal methods of analyses of the shells of noncanonical form]. Moscow, RUDN Publ., 540. (In Russ.)

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