from 01.10.2008 until now
Russian Federation
Making smooth outlines produced products may be due to the different requirements: aerodynamic, structural, aesthetic, etc. Therefore, the consideration of the subject of curves of the second order is included in many textbooks on descriptive geometry and engineering graphics. These curves can be used as a transition from one line to another with the first and to the secondorder smoothness. In the first part of this work dealt with the General rules of construction of second-order curves based on the properties of ticlid Dupin. Used the total for all of Konik the principle of creation — the center of the sphere that is in contact circumferentially with ciklidas the clock, when you change the radius of the sphere moves along the curve of the second order. The circle of contact of the sphere with Cichlidae Dupin always located in a plane passing through one of the two axes of cyclide, and each of these planes intersects cyclide by two circles of contact. This property formed the basis of graphical representation, common to all curves of the second order. Thus was created a unified principle of construction, implemented a systematic approach. In addition, the building has a connection with such a conversion as the Central homothety or the like. It considers the construction of an ellipse with different variants of the task: two focuses and a point belonging to the hyperbola; two values: a and b; by the actual diameter of the hyperbola is 2a and its tricks; the size 2b of the hyperbola and its foci; the vertices of the hyperbola and the point to her belonging; largest 2b hyperbola and the point, owned her; the equation of the hyperbola; the contour of cyclide Dupin. Also, eight options were considered the task of constructing the parabola.
pairing, descriptive geometry, cyclic surfaces, canal surface, cyclide of Dupin, the task of Apollonius.
Придание плавных очертаний изготовляемым изделиям может быть 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 [1–4; 6; 8; 9; 11; 13–16; 18; 25–28; 30]. Разработанные принципы построения кривых второго порядка были основаны на свойствах циклиды Дюпена [7; 10; 12; 19–24; 31].
В первой части данной работы [23] рассматривались общие правила построения кривых второго порядка на основе свойств циклид Дюпена. Применялся общий для всех коник принцип построения — центр сферы, соприкасающийся по окружности с циклидой Дюпена, при изменении радиуса этой сферы перемещается по кривой второго порядка. При этом окружность соприкосновения сферы с циклидой Дюпена всегда расположена в плоскости, проходящей через одну из двух осей циклиды, а каждая из этих плоскостей пересекает циклиду по двум окружностям соприкосновения. Это свойство легло в основу графического построения, общего для всех кривых второго порядка. Таким образом, был сформирован единый принцип построения, осуществлен системный подход. Кроме того, все рассматриваемые построения имеют связь с таким преобразованием как гомотетия (центральное подобие) [5; 17; 29].
1. Anisimov N.N., Kuznecov N.S., Kirillov A.F. Cherchenie i risovanie [The drawing and painting]. Moscow, Strojizdat Publ., 1983. 368 p. (in Russian).
2. Argunov B.I., Balk M.B. Geometricheskie postroenija na ploskosti [The geometric construction on the plane]. Moscow, Uchpedgiz Publ., 1957. 267 p. (in Russian).
3. Botvinnikov A.D., Vinogradov V.N., Vyshnepol´skij I.S. Cherchenie. 9 klass [Drawing. Grade 9]. Moscow, OOO «Izdatel´stvo AST» Publ., OOO «Izdatel´stvo Astrel´» Publ., 2015. 221 p. (in Russian).
4. Budasov B.V., Kaminskij V.P. Stroitel´noe cherchenie [Construction drawing]. Moscow, Strojizdat Publ., 1990. 464 p. (in Russian).
5. Vygodskij M.Ja. Analiticheskaja geometrija [Analytical geometry]. Moscow, Fizmatgiz Publ., 1963, 523 p. (in Russian).
6. Vyshnepol´skij I.S. Tehnicheskoe cherchenie [Technical drawing]. Moscow, Jurajt Publ., 2014. 319 p. (in Russian).
7. Gil´bert D., Kon-Fossen S. Nagljadnaja geometrija [Visual geometry]. Moscow, Leningrad, Ob´´edinionnoe nauchnotehnicheskoe izdatel´stvo NKTP SSSR, Glavnaja redakcija obshhe-tehnicheskoj literatury i nomografii Publ., 1936. 302 p. (in Russian).
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 Publ., 1951. (in Russian).
9. Zelenin E.V. Nachertatel´naja geometrija i cherchenie [Descriptive geometry and drawing]. Moscow, Gosudarstvennoe izdatel´stvo tehniko-teoreticheskoj literatury Publ., 1953. 524 p. (in Russian).
10. Klejn F. Vysshaja geometrija [Higher geometry]. Moscow, Leningrad, GONTI Publ., 1939. 400 p. (in Russian).
11. Koroev Ju.I. Cherchenie dlja stroitelej [Drawing for builders]. Moscow, KNORUS Publ., 1915. 256 p. (in Russian).
12. Krivoshapko S.N., Ivanov V.N. Enciklopediya analiticheskih poverhnostej [Encyclopedia of analytical surfaces]. Moscow, Knizhnyj dom «LIBROKOM» Publ., 2010. 560 p. (in Russian).
13. Levickij V.S. Mashinostroitel´noe cherchenie [Machine drawing]. Moscow, Vyssh. shk. Publ., 1988. 351 p. (in Russian).
14. Lomonosov G.G. Inzhenernaya grafika [Engineering graphics]. Moscow, Nedra Publ., 1984. 287 p. (in Russian).
15. Mihajlenko V.E., Ponomarev A.M. Inzhenernaya grafika [Engineering graphics]. Kiev, Vishha shkola Publ., 1980. 280 p. (in Russian).
16. Mogil´nyj I.M. Tehnicheskoe cherchenie [Technical drawing]. Kiev, Gostehizdat: Ukrainskoe otdelenie Publ., 1954 (in Russian).
17. Nadolinnyj V.A. Analiticheskie metody v konstruirovanii poverhnostej [Analytical methods in engineering surfaces]. Kiev, KPI Publ., 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"]. Sb. nauchno-metodicheskih statej po nachertatel´noj geometrii i inzh. grafike. Moscow, Vysshaja shkola Publ., 1985, V. 12, pp. 42-47. (in Russian).
19. Salkov N.A. Svojstva ciklidy Djupena i ih primenenie. Chast´ 1 [Properties Dupin Dupin and their application. Part 1]. Geometriya i grafika [Geometry and graphics], 2015, V. 3, I. 1, рр. 16-25. DOI:https://doi.org/10.12737/10454. (in Russian).
20. Salkov N.A. Svojstva ciklidy Djupena i ih primenenie. Chast´ 2 [Properties Dupin Dupin and their application. Part 2]. Geometriya i grafika [Geometry and graphics], 2015, V. 3, I. 2, рр. 9-23. DOI:https://doi.org/10.12737/12164. (in Russian).
21. Salkov N.A. Svojstva ciklidy Djupena i ih primenenie. Chast´ 3 [Properties Dupin Dupin and their application. Part 3]. Geometriya i grafika [Geometry and graphics], 2015, V. 3, I. 4, рр. 3-14. DOI:https://doi.org/10.12737/17345. (in Russian).
22. Salkov N.A. Svojstva ciklidy Djupena i ih primenenie. Chast´ 4 [Properties Dupin Dupin and their application. Part 4]. Geometriya i grafika [Geometry and graphics], 2016, V. 4, I. 1, рр. 21-32. DOI:https://doi.org/10.12737/17347. (in Russian).
23. Salkov N.A. Ciklidy Djupena i krivye vtorogo porjadka. Chast´ 1 [Dupin Cyclide and Second-Order Curves. Part 1]. Geometriya i grafika [Geometry and graphics], 2015, V. 4, I. 2, рр. 19-28. DOI: 10.12737/ (in Russian).
24. Salkov N.A. Jellips: kasatel´naja i normal´ [Ellipse: tangent and normal]. Geometriya i grafika [Geometry and graphics], 2013, V. 1, I. 1, pp. 35-37. DOI:https://doi.org/10.12737/2084. (in Russian).
25. 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, 264 р. (in Russian).
26. Simonin S.I. Inzhenerno-topograficheskoe cherchenie i nagljadnye izobrazhenija [Engineering and topographical drawing and descriptive image]. Moscow, Nedra Publ., 1979. 192 р. (in Russian).
27. Russkevich N.L., Tkach D.I., Tkach M.N. Spravochnik po inzhenerno-tehnicheskomu chercheniju [Reference engineering drawing]. Kiev, Budivel´nik Publ., 1980 (in Russian).
28. Godik E.I., Lysyanskij V.M., Mihaylenko V.E. Tehnicheskoe cherchenie [Technical drawing]. Kiev, Vishha shkola Publ., 1983. 440 р. (in Russian).
29. Chetveruhin N.F. Proektivnaja geometriya [Projective geometry]. Moscow, Uchpedgiz Publ., 1961. (in Russian).
30. Chumachenko G.V. Tehnicheskoe cherchenie [Technical drawing]. Moscow, KNORUS Publ., 2013. 296 р. (in Russian).
31. Dupin Ch. Développements de géometrié, P., 1813.
