Classification of analytical surfaces as applied to parametrical architecture and machine building

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At present time, the architects began to show interest in the creation of space large-span shapes due to advent of new materials and building construction technologies. Increased ergonomic requirements for structures and erections are now a great significance. This approach can be seen in many projects, both implemented in material and still being at the stage of drafts and blueprints. Now in economically safe countries, there has already been a splash in the creation of new unique structures, including spatial ones with complex surfaces. It is necessary to prepare for such prospect also in our country. Currently, a large number of surface classifications is offered, but they do not cover the all variety of known surfaces. This article introduces a classification of both new and widely known surface shapes as applied to the structures of building and industrial purposes. A detailed classification of analytic surfaces of 38 classes is constituted. An extensive bibliography containing 40 references is presented for convenient search of studies in the field of geometry and strength analysis of thin shells with the non-canonical middle surfaces.

About the authors

Elena A Grinko

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Head of Laborite, Assistant, Department of Civil Engineering, Engineering Academy


  1. Podgorniy A.L., Grinko E.A., Solovey N.A. Research of new forms of surfaces as applied to structures of divorce purposes. RUDN Journal of Engineering Researches, 2013, No. 1, 140—145 (In Russ.)
  2. Bulygin A.V. On one class of shells of reversed gauss curvature. Mekhanika tverdogo tela [Mechanics of Solid Body], 1977, No. 5, 97—104. (In Russ.)
  3. Krivoshapko S.N., Ivanov V.N. Encyclopedia of Analytical Surfaces. Springer International Publishing, Switzerland, 2015, 752.
  4. Krivoshapko S.N. Classification of ruled surfaces. Structural Mechanics of Engineering Constructions and Buildings, 2006, No. 1, 10—20. (In Russ.)
  5. Krivoshapko S.N., Ivanov V.N. Classification of cyclic surfaces. Structural Mechanics of Engineering Constructions and Buildings, 2006, No. 2, 25—34. (In Russ.)
  6. Mamieva I.A. On classification of analytical surfaces. In: International Scientific-and-Practical Conference “Engineering Systems — 2011”. Moscow: RUDN Publ., 2011, 63—65. (In Russ.)
  7. Belyakova M.S. The Increasing Effectiveness of Process of Structural-and-Technological designing on the base of development of informational system of surface modeling. PhD Diss. Moscow: MGTU Stankin Publ., 2007.
  8. Krivoshapko S.N., Bock Hyeng Ch.A. Classification of cyclic surfaces and geometrical research of canal surfaces. International Journal of Research and Reviews in Applied Sciences, 2012, Vol. 12, Iss. 3, 360—374.
  9. Krivoshapko S.N., Shambina S.L. Surfaces of congruent sections of the pendulum type on a circular cylinder. Geometricheskoe modelirovanie i komp’yuternie tehnologii: teoriya, praktika, obrazovanie. Proc. VI Int. Scientifical-and-Pract. Conf., Ukraine, Kharkov, April 21—24, 2009. Kharkov: KhGUPT, 2009, 34—39. (In Russ.)
  10. Krivoshapko S.N., Shambina S.L. Surfaces of velaroidal type on ring plan with two families of sinusoids. Prazi Tavriyskogo Derzhavnogo Agrotekhnol. Univer.: XI Int. Scient.-and-Pract. Conf. “Suchasni problemy heometrychnoho modeliuvannia SPHM-11”, 9—12 June 2009, Melitopol, Ukraine. 2009. 29—33. (In Russ.)
  11. Nassar H., Abdel-All R.A. Hussien and Taha Youssef. Hasimoto surfaces. Life Science Journal, 2012, 9(3), 556—560.
  12. Sachs H. Einige Kennzeichnungen der Edlinger — Flächen. Monatsh. Math., 1973, 77(3), 241— 250.
  13. Lalan V. Les formes minima des surfaces d’Ossian Bonnet. Bull. Soc. Math., 1949, 77, 102—127.
  14. Degen W. Die zweifachen Blutelschen Kegelschnittflächen. Manuscr. math., 1986, 55(1), 9—38.
  15. López R. Special Weingarten surfaces foliated by circles. Monatsh Math., 2008, 154, 289—302.
  16. Grinko E.A. Survey works on geometry, durability, stability, dynamics, and application of environments with median surfaces of various classes. Montazhn. i spets. raboty v stroitelstve, 2012, No. 2, 15—21 (in Russ.)
  17. Krivoshapko S.N. Analysis and design of helical structures used in building and building machines: review. Moscow: Rosstroy Rossii, VNIINTPI Publ., 2006, Vol. 1—2, Series Building Structures and Materials, 68. (In Russ.)
  18. Lebedev V.A. Thin-Walled Umbrella Shells. Leningrad: Gosstoyizdat Publ., 1958, 172. (In Russ.)
  19. Krivoshapko S.N. Geometrical investigations of umbrella-type surfaces. Structural Mechanics of Engineering Constructions and Buildings, 2005, No. 1,11—17(In Russ.)
  20. Miftakhutdinov I.H. Visual Geometry of Shells with Minimal Middle Surface. Kazan: Novoe Znanie Publ., 2009, 40. (In Russ.)
  21. Krivoshapko S.N. The application of conoid and cylindroid in forming of buildings and structures of shell type. Building and Reconstruction, 2017, No. 5(73), 34—44. (In Russ.)
  22. Starostin E.L., Van Der Heijden G.H.M. The shape of a Möbius strip. Nat. Mater., 2007, 6, 563—567.
  23. Nitsche J.C.C. Cyclic surfaces of constant mean curvature. Nachr. Akad. Wiss. Gottingen Math. Phys., 1989, II, 1, 1—5.
  24. Maximov V.I. (ed.). The Illustrated Concise Dictionary of Russian Scientific and Technical Terms. Moscow: Russky Yazyk Publ., 1994, 379. (In Russ.)
  25. Ivanov V.N., Shmeleva A.A. Geometry and formation of the thin-walled space shell structures on the base of normal cyclic surfaces. Structural Mechanics of Engineering Constructions and Buildings, 2016, No. 6, 3—8. (In Russ.)
  26. Ivanov V.N. Geometry and forming of the polyhedral box type surfaces on base cyclic surface. Structural Mechanics of Engineering Constructions and Buildings, 2012, No. 2, 3—10. (In Russ.)
  27. Bock Hyeng Ch.A., Yamb E.B. Application of cyclic shells in architecture, machine design, and bionics. Int. J. of Modern Engineering Research, 2012, 2(3), 799—806.
  28. Shulikovskiy V.I. Classical Differential Geometry. Moscow: Fizmatgiz Publ., 1963, 379. (In Russ.)
  29. Martirosov A.L. Rotative Transformation of Space. Rostov-na-Donu, 2006, 248. (In Russ.)
  30. Mamieva I.A., Razin A.D. Parametrical architecture in Moscow. Arkhitectura i stroitelstvo Rossii, 2014, No. 6, 24—29. (In Russ.)
  31. Virchenko G.A., Shambina S.L. Computer variant surface modeling of bionic architectural forms. RUDN Journal of Engineering Researches, 2016, No. 3, 79—83. (In Russ.)
  32. Saleh M.S. Application of modern techniques aided design for forming and calculation of erections of progressive architecture. Structural Mechanics of Engineering Constructions and Buildings, 2016, No. 6, 8—13. (In Russ.)
  33. Ivanov V.N., Romanova V.A. Constructive Forms of Spatial Structures (Visualization of Surfaces in MathCad, AutoCad). Moscow: ASV Publ., 2016, 412. (In Russ.)
  34. Radzevich S.P. Generation of Surfaces: Kinematic Geometry of Surface Machining. 1st Edition. CRC Press, 2017, 738.
  35. Ivanov V.N., Valensya Rodriges E.G. Thin-walled shell structures on the base of Joachimsthal canal surfaces. Structural Mechanics of Engineering Constructions and Buildings, 2016, No. 2, 15—21. (In Russ.)
  36. Cheshkova M.A. Clifford’s torus and Klein’s bottle. Izvestiya Altayskogo Gosudarstvennogo Univ., 2017, No. 1(93), 144—147. (In Russ.)
  37. Zolotukhin V.F. Classification of surfaces. Volgograd. Politekh. In-t, 1979, 11 p., Ruk. dep. v VINITI March 17, 1980, № 1007-89 Dep.
  38. Murre J.P. Classification of algebraic varieties. Nieuw. Arch. Wisk., 1977, 25(3), 308—338.
  39. Bazderova T.A. Problems of classification of ruled surfaces and design of their devel-opments. Kuzbas. Politekh. In-t, Kemerovo, 1985, 53 p., Ruk. dep. v VINITI June 19, 1985, № 4375-85 Dep.
  40. Druzhinskiy I.A. Slozhnie Poverkhnosti [Complex Surfaces: Mathematical Description and Technological Description]. Leningrad: 1985, 263 p.



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