The surface of non-linear rotation

The paper considers a geometric scheme, a mathematical model and an algorithm for shaping a non-linear rotation surface. It is known that in Euclidean geometry and mechanics the transformation of rotation is linear, while distance and angle are its invariants. The authors proposed a geometric scheme of non-linear rotation, in which the axis of rotation is a smooth spatial curve and the object of rotation is a smooth line. Several propositions, a lemma and a theorem are proved, which allow one to form the initial data in the problem of nonlinear rotation, the solution of which is the parametric equations of smooth surfaces. The research results make it possible to expand the variety of cyclic surfaces in the existing classification of analytic surfaces. They can also be useful in the creation of CAD, which provides for the design of surface forms of products for mechanical engineering, construction, architecture and other practical areas based on cyclic surfaces.

1. Grigor'yev M. I. Postroyeniye obobshchennykh poverkhnostey vrashcheniya [Construction of generalised rotation surfaces] // Seminar «DNA&CAGD». Izbrannyye doklady. Seminar «DNA&CAGD». Izbrannyye Doklady. 2007. P. 1-7. URL: http://dha.spb.ru/PDF/GeneralizedRevolution.pdf (accessed: 27.02.2023) (In Russ.).

2. Grigor'yev M. I., Malozemov V. N. Sostavnyye krivyye i poverkhnosti Bez'ye. Analiticheskiy podkhod [Compound curves and Büzier surfaces. The analytical approach]. Lambert Academic Publishing, 2010. 132 p. ISBN 978-3-8433-0323-1. (In Russ.).

3. Beglov I. A., Rustamyan V. V. Metod vrashcheniya geometricheskikh ob”yektov vokrug krivolineynoy osi [Method of rotation of geometrical objects around the curvilinear axis] // Geometriya i grafika. Geometry and Graphics. 2017. No. 3. P. 45-50. DOI: 10.12737/article_59bfa4eb0bf488.99866490. EDN: ZGWEMP. (In Russ.).

4. Beglov I. A. Computer geometric modeling of quasirotation surfaces // Journal of Physics: Conference Series. 2021. Vol. 1901. P. 012057. DOI: 10.1088/1742-6596/1901/1/012057. (In Engl.).

5. Osipov V. A. Mashinnyye metody proyektirovaniya nepreryvnokarkasnykh poverkhnostey [Machine-assisted design methods for continuous frame surfaces]. Moscow, 1979. 248 p. (In Russ.).

6. Osipov V. A., Osipova L. I. Teoreticheskiye osnovy karkasno-kinematicheskogo metoda napravlyayushchey linii [Theoretical foundations of the frame-kinematic guide line method] // Izvestiya vysshikh uchebnykh zavedeniy. Aviatsionnaya tekhnika. Izvestiya Vysshikh Uchebnykh Zavedenii. Aviatsionnaya Tekhnika. 1980. No. 4. P. 48-53. (In Russ.).

7. Markin L. V., Korn G. V., Kui M. Kh. [et al.]. Diskretnyye modeli geometricheskogo modelirovaniya komponovki aviatsionnoy tekhniki [Discrete models of aircraft equipment layout geometric modeling] // Trudy MAI. Proceedings of MAI. 2016. No. 86. 16 p. EDN: VUDSTD. (In Russ.).

8. Khtun N. N. Razrabotka i issledovaniye retseptornykh geometricheskikh modeley telesnoy trassirovki [Development and research of receptive geometric models of body tracing]. Moscow, 2014. 26 p. (In Russ.).

9. Blyashke V. Differentsial'naya geometriya I geometricheskiye osnovy teorii otnositel'nosti Eynshteyna. V 2 t. T. 1. Elementarnaya differentsial'naya geometriya. [Differential geometry and the geometrical foundations of Einstein's theory of relativity. In 2 vols. Vol. 1. Elementary differential geometry]. Moscow; Leningrad, 1935. 330 p. (In Russ.).

10. Kartan E. Teoriya konechnykh nepreryvnykh grupp i differentsial'naya geometriya, izlozhennyye metodom podvizhnogo repera [Theory of finite and continuous groups and differential geometry treated by the moving frame method] / trans. from. fr. S. P. Finikova. Moscow, 1998. 368 p. ISBN 5-80100-297-9. (In Russ.).

11. Zeyliger D. N. Kompleksnaya lineychataya geometriya [Complex linear geometry]. Moscow; Leningrad, 1934. 196 p. (In Russ.).

12. Yakubovskiy A. M. Nekotoryye voprosy konstruirovaniya poverkhnostey s pomoshch'yu trekhgrannika Frene [Some issues of surface construction with the Fresnier trihedron] // Trudy un-ta Druzhby Narodov im. P. Lumumby. Proceedings of the P. Lumumba Peoples' Friendship University. Moscow, 1967. Vol. 26. P. 23-32. (In Russ.).

13. Panchuk K. L., Niteyskiy A. S. Contact of the Ruled Nondevelopable Surfaces // Proceedings of the 16th International Conference on Geometry and Graphics, 4-8 August 2014. Innsbruck: University Press, 2014. Р. 216-223. (In Engl.).

14. Niteyskii A. S. Konstruirovaniye torsovoy poverkhnosti metodom podvizhnogo trekhgrannika Frene [Design of ruled surfaces by moving Frenet trihedron] // Omskiy nauchnyy vestnik. Omsk Scientific Bulletin. 2013. No. 2 (120). P. 151-153. EDN: RNEIEX. (In Russ.).

15. Korchagin D. S., Panchuk K. L. Metod geometro- dinamicheskogo formoobrazovaniya lineychatykh polos [Methods of geometry-dynamic shaping linear strips] // Vestnik KuzGTU. Bulletin of the Kuzbass State Technical University. 2013. Issue 6 (100). P. 89-92. EDN: RUDWXJ. (In Russ.).

16. Korchagin D. S., Panchuk K. L. Forming of the Spline Similar Linear Strip // Proceedings of the 16th International Conference on Geometry and Graphics. Innsbruck, Austria. 2014. P. 428-436. (In Engl.).

17. Krivoshapko S. N. Entsiklopediya analiticheskikh poverkhnostey: boleye 500 poverkhnostey, 38 klassov: matematikam, inzheneram, arkhitektoram [Encyclopedia of analytical surfaces: more than 500 surfaces, 38 classes: mathematicians, engineers, architects]. Moscow, 2010. 556 p. ISBN 978-5-397-00985-0. (In Russ.).

18. Krivoshapko S. N., Ivanov V. N. Klassifikatsiya tsiklicheskikh poverkhnostey [Classification of cyclic surfaces] // Stroitel'naya Mekhanika Inzhenernykh Konstruktsiy i Sooruzheniy. Structural Mechanics of Engineering Structures and Facilities. 2006. No 2. P. 25-34. (In Russ.).

19. Krivoshapko S., Hyeng C. Geometrical research of rare types of cyclic surfaces // International Journal of Research and Reviews in Applied Sciences. 2012. Vol. 12, no. 3. Р. 346-359. DOI: 10.1088/1742-6596/1901/1/012057. (In Engl.).

20. Bock Hyeng Ch. A., Yamb E. B. Application of Cyclic Shells in Architecture, Machine Design, and Bionics // International Journal of Modern Engineering Research. 2012. Vol. 2, no. 3. P. 799-806. (In Engl.).

21. Ivanov V. N., Shmeleva A. A. Geometriya I formoobrazovaniye tonkostennykh prostranstvennykh konstruktsiy na osnove normal'nykh tsiklicheskikh poverkhnostey [Geometry and formation of the thin-walled space shell structures on the base of normal cyclic surfaces] // Stroitel'naya Mekhanika Inzhenernykh Konstruktsiy i Sooruzheniy. Structural Mechanics of Engineering Structures and Structures. 2016. No 6. P. 3-8. EDN: WYOLRL. (In Russ.).

22. Ivanov V. N., Rynkovskaya M. I. Primeneniye tsiklicheskikh poverkhnostey v arkhitekture zdaniy, konstruktsiy i izdeliy [Application of circular surfaces to the architecture of the buildings, structures and products] // Vestnik Rossiyskogo universiteta druzhby narodov. Ser. Inzhenernyye issledovaniya. Rudn Journal of Engineering Research. 2015. No 3. P. 111-119. EDN: ULXYJH. (In Russ.).