from 01.10.2008 until now
Russian Federation
Making smooth shapes of various products is caused by the following requirements: aerodynamic, structural, aesthetic, etc. That’s why the review of the topic of second-order curves is included in many textbooks on descriptive geometry and engineering graphics. These curves can be used as a transition from the one line to another as the first and second order smoothness. Unfortunately, in modern textbooks on engineering graphics the building of Konik is not given. Despite the fact that all the second-order curves are banded by a single analytical equation, geometrically they unites by the affiliation of the quadric, projective unites by the commonality of their construction, in the academic literature for each of these curves is offered its own individual plot. Considering the patterns associated with Dupin cyclide, you can pay attention to the following peculiarity: the center of the sphere that is in contact circumferentially with Dupin cyclide, by changing the radius of the sphere moves along the second-order curve. The circle of contact of the sphere with Dupin cyclide is always located in a plane passing through one of the two axes, and each of these planes intersects cyclide by two circles. This property formed the basis of the graphical constructions that are common to all second-order curves. In addition, considered building has a connection with such transformation as the dilation or the central similarity. This article considers the methods of constructing of second-order curves, which are the lines of centers tangent of the spheres, applies a systematic approach.
pairing, descriptive geometry, engineering graphics, cyclic surfaces, canal surface, Dupin cyclide.
1. Общие положения
Придание плавных очертаний изделиям обусловлено различными требованиями: аэродинамическими, прочностными, эстетическими и т.д. Поэтому рассмотрение темы кривых второго порядка входит во многие учебники по начертательной геометрии и инженерной графике. Эти кривые могут использоваться как переходные [18] от одной линии к другой.
К сожалению, в современных учебниках по инженерной графике не всегда даются построения коник в полном объеме. Так, например, в [1] присутствует только пояснение, что кривые второго порядка имеют место быть при сечении квадрики плоскостью, но ни одного построения не предлагается. Такое положение дел можно принять для учебника [3], написанного для школьников, но никак не для учебника для вуза. В работах [6; 29], задекларированных как учебники для технических направлений обучения, среди лекальных кривых даются только номинальные сведения об эллипсе. В учебнике для строителей [4; 11] есть не только пояснения, но предлагается и по одному-два из самых известных построений коник. Тут же возникает вопрос: почему для строителей построение, хотя бы одно, есть, а для механиков отсутствует? А вот в справочнике [26] о кониках ничего не сказано.
1. Anisimov N.N., Kuznecov N.S., Kirillov A.F. Cherchenie i risovanie [The drawing and painting]. Moscow, Strojizdat Publ., 1983. (in Russian).
2. Argunov B.I., Balk M.B. Geometricheskie postroenija na ploskosti [Geometric constructions on the plane]. Moscow, Uchpedgiz Publ., 1957. (in Russian).
3. Botvinnikov A.D., Vinogradov V.N., Vyshnepol´skij I.S. Cherchenie. 9 klass [Drawing. Grade 9]. Moscow, AST Publ., Astrel´ Publ., 2015.
4. Budasov B.V., Kaminskij V.P. Stroitel´noe cherchenie [Construction drawing]. Moscow, Strojizdat Publ., 1990. (in Russian).
5. Vygodskij M.Ja. Analiticheskaja geometrija [Analytical geometry]. Moscow, Fizmatgiz Publ., 1963. (in Russian).
6. Vyshnepol´skij I.S. Tehnicheskoe cherchenie [Technical drawing]. Moscow, Jurajt Publ., 2014. (in Russian).
7. Gil´bert D., Kon-Fossen S. Nagljadnaja geometrija [Visual geometry]. Moscow, Leningrad, Obyedinennoe nauchno-tehnicheskoe izdatel´stvo NKTP SSSR, Glavnaja redakcija obshhetehnicheskoj literatury i nomografii Publ., 1936.
8. Gordon V.O. Cherchenie dlja ploskih i prostranstvennyh figur [For drawing of plane and solid figures]. Moscow, Gosudarstvennoe uchebno-pedagogicheskoe izdatel´stvo ministerstva prosveshhenija RSFSR, 1951. (in Russian).
9. Zelenin E.V. Nachertatel´naja geometrija i cherchenie [Descriptive geometry and drawing]. Moscow, Gosudarstvennoe izdatel´stvo tehniko-teoreticheskoj literatury, 1953. (in Russian).
10. Klein F. Vysshaja geometrija [Higher geometry]. Moscow, Leningrad, GONTI Publ., 1939.
11. Koroev Ju.I. Cherchenie dlja stroitelej [Drawing for builders]. Moscow, KNORUS Publ., 1915. (in Russian).
12. Krivoshapko S.N., Ivanov V.N. Jenciklopedija analiticheskih poverhnostej [Encyclopedia of analytical surfaces]. Moscow, LIBROKOM Publ., 2010. (in Russian).
13. Levickij V.S. Mashinostroitel´noe cherchenie: Uchebnik dlja studentov vtuzov [Machine drawing]. Moscow, Vysshaya shkola Publ., 1988. (in Russian).
14. Lomonosov G.G. Inzhenernaja grafika [Engineering graphics]. Moscow, Nedra Publ., 1984. (in Russian).
15. Mihajlenko V.E., Ponomarev A.M. Inzhenernaja grafika [Engineering graphics]. Kiev, Vishha shkola Publ., 1980. (in Russian).
16. Mogil´nyj I.M. Tehnicheskoe cherchenie [Technical drawing]. Kiev, Gostehizdat: Ukrainskoe otdelenie, 1954. (in Russian).
17. Nadolinnyj V.A. Analiticheskie metody v konstruirovanii poverhnostej [Analytical methods in engineering surfaces]. Kiev, KPI, 1981. (in Russian).
18. Salkov N.A. O racional´nom graficheskom reshenii zadach po teme "Soprjazhenija" [About rational graphical solution of problems on the subject of "Pairing"]. Moscow, Vysshaja shkola Publ., 1985, I. 12, pp. 42-47. (in Russian).
19. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast´ 1 [Properties cyclid of Dupin and their application. Part 1]. Geometrija i grafika [Geometry and graphics]. 2015, V. 3, I. 1, pp. 16-25. DOI:https://doi.org/10.12737/10454. (in Russian).
20. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast´ 2 [Properties cyclid of Dupin and their application. Part 2]. Geometrija i grafika [Geometry and graphics]. 2015, V. 3, I. 2, pp. 9-23. DOI:https://doi.org/10.12737/12164. (in Russian).
21. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast´ 3 [Properties cyclid of Dupin and their application. Part 3]. Geometrija i grafika [Geometry and graphics]. 2015, V. 3, I. 4, pp. 3-14. DOI:https://doi.org/10.12737/17345. (in Russian).
22. Salkov N.A. Svojstva ciklid Djupena i ih primenenie. Chast´ 4 [Properties cyclid of Dupin and their application. Part 4]. Geometrija i grafika [Geometry and graphics]. 2016, V. 4, I. 1, pp. 21-32. DOI:https://doi.org/10.12737/17347. (in Russian).
23. Sal´kov N.A. Jellips: kasatel´naja i normal´ [Ellipse: tangent and normal]. Geometrija i grafika [Geometry and graphics]. 2013, V. 1, I. 1, pp. 35-37. DOI:https://doi.org/10.12737/2084. (in Russian).
24. Sberegaev N.P., Gerb M.A. Kratkij spravochnik po nachertatel´noj geometrii i mashinostroitel´nomu chercheniju [A brief guide on descriptive geometry and engineering drawing]. Moscow, Leningrad, Mashinostroenie Publ., 1965. (in Russian).
25. Simonin S.I. Inzhenerno-topograficheskoe cherchenie i nagljadnye izobrazhenija [Engineering and topographical drawing and descriptive image]. Moscow, Nedra Publ., 1979. (in Russian).
26. Russkevich N.L., Tkach D.I., Tkach M.N. Spravochnik po inzhenerno-tehnicheskomu chercheniju [Reference engineering drawing]. Kiev, Budivel´nik Publ., 1980. (in Russian).
27. Godik E.I., Lysjanskij V.M., Mihajlenko V.E., Ponomarev A.M. Tehnicheskoe cherchenie [Technical drawing]. Kiev, Vishha shkola Publ., 1983. (in Russian).
28. Chetveruhin N.F. Proektivnaja geometrija [Projective geometry]. Moscow, Uchpedgiz Publ., 1961. (in Russian).
29. Chumachenko G.V. Tehnicheskoe cherchenie [Technical drawing]. Moscow, KNORUS Publ., 2013. (in Russian).
30. Dupin Ch. Développements de géometrié. P., 1813.
