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In the first part of this paper has discussed about the basic properties of cyclide Dupin, and has gave some examples of their applications: three ways of solving the problem of Apollonius exclusively by means of compass and ruler, using identified properties cyclide Dupin, that is, given a classical solution of the problem. In the second of part of the work continued consideration of the properties of cyclide Dupin. Proposed and proved the possibility ask cyclide Dupin arbitrary ellipse as the center line of the forming a plurality of spheres and a sphere with the center belong - ing to the ellipse. Proved the adequacy of this information is used to build the cyclide Dupin. Geometrically proved that the focal line of cychlid are not that other, as curves of the second order. Given the graphical representation of the focal lines of cychlid. Shown polyconic compliance focal lines of cichlid of Dupin, which is considered in all four cases. The proposed formation of the hyperbolic surfaces of the fourth order with one or two primary curves of the second order, in this case ellipses. Apply sofocus this ellipse the hyperbola. Although the primary focus of the ellipse lying in the plane of the hoe, with the center coinciding with the origin of coordinates, is stationary, and the coordinate system rotates around the z axis. Then the points of intersection of the rotating coordinates x and y with a fixed ellipse specify new values for the major and minor axis of the ellipse with resultant changes in the form defocuses of the hyperbola. Although this modeling is not directly connected with Cychlidae Dupin, but clearly follows from the properties of its focal curves – curves of the second order. Withdrawn Equations of the surface and its throat.
descriptive geometry, circular surface, Kanaloa surface, a Dupin cyclide, the problem of Apollonius, the task Farm.
В первой части предлагаемой работы [20] рассматривался вопрос об основных свойствах циклиды Дюпена [5; 6; 8; 10–12; 23], а также приводились некоторые примеры их применения: три способа решения задачи Аполлония [9] исключительно при помощи циркуля и линейки, используя выявленные свойства циклиды, т.е. показывалось классическое решение задачи. Во второй части работы продолжено рассмотрение свойств циклиды Дюпена. При исследовании были применены методы аналитической [3], проективной [21] и элементарной [1; 2] геометрий. Методам аналитической геометрии в статьях придается особое значении [5; 6; 8; 10–18; 23], поскольку они позволяют получить математически точное решение.
1. Argunov B.I., Balk M.B. Geometricheskie postroenija na ploskosti [Geometric constructions on the plane]. Moscow, Uchpedgiz Publ., 1957. (in Russian).
2. Berzhe M. Geometrija [The geometry]. V. 1-2. Moscow, Mir Publ., 1984.
3. Vygodskij M.Ja. Analiticheskaja geometrija [Analytical geometry]. Moscow, Fizmatgiz Publ., 1963. 523 p. (in Russian).
4. Gil´bert D., Kon-Fossen S. Nagljadnaja geometrija [Visual geometry]. Moscow, Leningrad, Obyedinennoe nauchnotehnicheskoe izdatel´stvo NKTP SSSR, Glavnaja redakcija obshhetehnicheskoj literatury i nomografii Publ., 1936.
5. Grjaznov Ja.A. Otsek kanalovoj poverhnosti kak obraz cilindra v rasslojaemom obrazovanii [Bay canal surface as a cylinder in rassloennom education]. Geometrija i grafika [Geometry and graphics], 2012, v. 1, i. 1, pp. 17-19. DOI:https://doi.org/10.12737/2077. (in Russian).
6. Ivanov G.S. Konstruktivnyj sposob issledovanija cvojstv parametricheski zadannyh krivyh [Constructive way to study the properties of parametrically defined curves]. Geometrija i grafika [Geometry and graphics], 2012, v. 2, i. 3, pp. 3-6. DOI:https://doi.org/10.12737/6518. (in Russian).
7. Klein F. Vysshaja geometrija [Higher geometry]. Moscow, Leningrad, GONTI Publ., 1939.
8. Krivoshapko S.N., Ivanov V.N. Enciklopedija analiticheskih poverhnostej [Encyclopedia of analytical surfaces]. Moscow, LIBROKOM Publ., 2010. (in Russian).
9. Levickij V.S. O teme «Soprjazhenija» v kurse «Inzhenernaja grafika» [About "Mates" in the course "Engineering graphics"]. Sbornik nauchno-metodicheskih statej po nachertatel´noj geometrii i inzhenernoj grafike [Collection of scientific and methodological articles on descriptive geometry and engineering graphics]. Moscow, Vysshaja shkola Publ., 1980, pp. 44-51. (in Russian).
10. Nadolinnyj V.A. Analiticheskie metody v konstruirovanii poverhnostej [Analytical methods in the design of surfaces]. Kiev, KPI Publ., 1981.
11. Salkov N.A. O nekotoryh zakonomernostjah, imejushhih mesto pri kasanii sfer [Some of the regularities that occur when the spheres touch]. Prikl. geometrija i inzh. grafika [Applied Geometry and Engineering Graphics]. Kiev, Budivel´nik Publ., 1981, i. 32, pp. 113-115. (in Russian).
12. Salkov N.A. Ob osobennostjah osi torovoj poverhnosti peremennogo radiusa [About the features of the axis of the torus sleeve surface of variable radius]. Prikl. geometrija i inzh. grafika. [Applied Geometry and Engineering Graphics]. Kiev, Budivel´nik Publ., 1982, i. 33, pp. 79-80. (in Russian).
13. Salkov N.A. Geometricheskie parametry grohota [The geometrical parameters of the rumble]. Prikl. geometrija i inzh. Grafika [Applied Geometry and Engineering Graphics]. Kiev, Budivel´nik Publ., 1987, i. 43, pp. 69-71. (in Russian).
14. Salkov N.A. Geometricheskoe i matematicheskoe modelirovanie virazhnyh uchastkov avtomobil´nyh dorog [Geometric and mathematical modeling of curved sections of roads]. Trudy MADI: Vychislitel´naja geometrija i mashinnaja grafika v zadachah SAPR avtomobilestroenija i avtomobil´nyh dorog [Proceedings MADI: Computational geometry and computer graphics in the CAD tasks automobile and highways]. Moscow, 1989, pp. 4-9. (in Russian).
15. Salkov N.A. Geometricheskoe i programmno-matematicheskoe modelirovanie linejnyh i poverhnostnyh form avtomobil´nyh dorog. Kand. Diss. [Geometric and mathematical modelling of linear and superficial forms of roads. Cand. Diss.]. Moscow, MADI Publ., 1990. (in Russian).
16. Salkov N.A. Analiticheskoe predstavlenie proektnyh gorizontalej poverhnostnyh form avtomobil´nyh dorog [Analytical representation of design contours to surface shapes of roads]. Aktual´nye problemy gradostroitel´stva i zhilishhno-kommunal´nogo kompleksa. Mezhdunarodnaja nauchno-prakticheskaja konferencija 15-16 maja 2003 g. [Actual problems of urban development and housing and communal services. International scientific-practical conference on 15-16 May 2003]. Moscow, MIKHiS Publ., 2003, pp. 61-62. (in Russian).
17. Salkov N.A. Kinematicheskoe sootvetstvie vrashhajushhihsja prostranstv [Kinematic compliance of rotating spaces]. Geometrija i grafika [Geometry and graphics], 2013, v. 1, i. 1, pp. 4-10.https://doi.org/10.12737/2074. (in Russian).
18. Salkov N.A. Grafo-analiticheskoe reshenie nekotoryh chastnyh zadach kvadratichnogo programmirovanija [Graph-analytic Solution of Some Special Problems of Quadratic Programming]. Geometrija i grafika [Geometry and graphics]. 2014, v. 2, i. 1, pp. 3-8. DOI:https://doi.org/10.12737/3842. (in Russian).
19. Salkov N.A. Parametricheskaja geometrija v geometricheskom modelirovanii [Parametric Geometry in Geometric Modeling]. Geometrija i grafika [Geometry and graphics], 2014, v. 2, i. 3, pp. 7-13. DOI:https://doi.org/10.12737/6519. (in Russian).
20. Salkov N.A. Ciklidy Djupena i ih primenenie. Chast´ 1 [Properties of Cyclide Dyupen 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).
21. Chetveruhin N.F. Proektivnaja geometrija [Projective geometry]. Moscow, Uchpedgiz Publ., 1961. (in Russian).
22. Enciklopedija jelementarnoj matematiki. Kniga chetvertaja - Geometrija [Encyclopaedia of elementary mathematics. Book four]. Moscow, Nauka Publ., 1966. (in Russian).
23. Jakubovskij A.M. Issledovanija analiticheskogo metoda zadanija ciklid Djupena pri vyjavlenii ih iz kongrujencij okruzhnostej. Trudy UDN. V. 53, i. 4, Prikladnaja geometrija. Moscow, 1971, pp. 26-40. (in Russian).
24. Dupin Ch. Développements de géometrié, Paris, 1813.
