Russian Federation
Has been presented a geometrical proof of a theorem stating that when a plane section crosses second-order revolution surfaces (rotation quadrics, RQ), such types of conics as ellipse, hyperbola or parabola are formed. The theorem amplifies historically famous Dandelin theorem, which provides geometric proof only for the circular cone, and extends the proof to all RQ: ellipsoid, hyperboloid, paraboloid and cylinder. That is why the theorem described below has been called as Generalized Dandelin theorem (GDT). The GDT proof has been constructed on a little-known generalized definition (GDD) of the conic. This GDD defines the conic as a line, that is a geometrical locus of points (GLP) P, for which ratio q = PT / PD = const, where PT is tangential distance from the point to the circle inscribed in the line, and PD is distance from the point to the straight line passing through the tangency points of the circle and the line. Has been presented a proof of GDD for all types of conics as their necessary and sufficient condition. The proof is in the construction of a circular cone and inscribed in sphere which is tangent to a cutting plane line at two points. For this construction is defined the position of a cutting plane, giving in section the specified conic. On the GDD basis has been proved the GDT for all the RQ with the arbitrary position of the cutting plane. For the proving a tangent sphere is placed in the quadric. An auxiliary cutting plane passing through the quadric axis is introduced. It is proved that in a section by axial plane the GDD is performed as a necessary condition for the conic. The relationship between the axial section and the given one is established. This permits to make a conclusion that in the given section the GDD is performed as the conic’s sufficient condition. Visual stereometrical constructions that are necessary for the theorem proof have been presented. The implementation of constructions using 3D computer methods has been considered. The examples of constructions in AutoCAD package have been demonstrated. Some constructions have been carried out with implementation of 2D parameterization. With regard to affine transformations the possibility for application of Generalized Dandelin theorem to all elliptic quadrics has been demonstrated. This paper is meant for including the GDT in a new training course on theoretical basis for 3D engineering computer graphics as a part of students’ geometrical-graphic training.
Dandelin theorem, Dandelin structure, Dandelin spheres, conic sections, quadrics, second-order surfaces, degree of point, parameterization, AutoCAD.
Введение
Конические сечения, их построение и исследование являются разделом учебного курса начертательной геометрии (НГ) [6; 9; 13; 23]. Основу этого раздела составляет теорема Данделена, доказывающая возникновение кривых второго порядка — коник:
эллипса, гиперболы, параболы — при сечении прямого кругового конуса плоскостью.
1. Adamar Zh. Jelementarnaja geometrija [Elementary geometry]. Planimetriya [Planimetry]. Moscow, Uch. Ped. Publ., 1948. 608 p. (in Russian).
2. Adamar Zh. Jelementarnaja geometrija [Elementary geometry]. Stereometrija [Stereometry]. Moscow, Uch. Ped. Publ., 1951. 760 p. (in Russian).
3. Akopyan A.V., Zaslavskiy A.A. Geometricheskie svoystva krivyih vtorogo poryadka [Geometric properties of secondorder curves]. Moscow, MTsNMO Publ., 2007. 136 p. (in Russian).
4. Gilbert D., Kon-Fossen S. Naglyadnaya geometriya [Visual geometry]. Moscow, Nauka Publ., 1981. 344 p. (in Russian).
5. Girsh A.G. Nachala kompleksnoy geometrii. Izbrannyie zadachi konstruk-tivnoy geometrii s resheniyami. Chast II — 3D [Beginning complex geometry. Selected problems of constructive geometry-tive solutions. Part II — 3D]. Kassel Publ., 2013. 100 p. (in Russian).
6. Gordon V.O., Semencov-Ogievskij M.A. Kurs nachertatel'noj geometrii [Course descriptive geometry]. Moscow, Vyissh. shk. Publ., 2008. 270 p. (in Russian)
7. Kheifetc A.L, Loginovskiy A.N., Butorina I.V., Vasileva V.N. Inzhenernaya 3D-kompyuternaya grafika [Engineering 3D computer graphics]. Moscow, “Yurayt” Publ., 2015. 602 p. (in Russian)
8. Korotkiy V.A. Dvoynoe prikosnovenie v puchke poverhnostey vtorogo poryadka [Double-Tap in a Beam of Second Order Surfaces]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 1, pp. 9–14. (in Russian). DOI: 10.12737/3843
9. Kryilov N.N., Ikonnikova G.S., Nikolaev V.L., Lavruhina N.M. Nachertatelnaya geometriya [Descriptive geometry]. Moscow, Vyissh. shk. Publ., 2010. 224 p. (in Russian)
10. Loginovskiy A.N., Kheyfets A.L. Reshenie zadach na osnove parametrizatsii v pakete AutoCAD [Tasks decision on parameterization basis in AutoCAD package]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 2, pp. 58–62. (in Russian). DOI: 10.12737/793
11. Muskhelishvili N.I. Kurs analiticheskoy geometrii [Analytical Geometry Course]. Moscow, Vysshaya shkola Publ., 1967. 656 p. (in Russian)
12. Nilov F.K. Obobschennoe opredelenie konik [A generalized definition of a conic]. 2015. Available at: https://www.youtube. com/watch?v=KYobKNvp1gI (in Russian)
13. Peklich V.A. Nachertatelnaya geometriya [Descriptive geometry]. Moscow, ASV Publ., 2007. 272 p. (in Russian)
14. Poya D. Matematika i pravdopodobnyie rassuzhdeniya [Mathematics and plausible reasoning]. Moscow, Nauka Publ., 1975. 464 p. (in Russian)
15. Postnikov M.M. Analiticheskaya geometriya [Analytic geometry]. Moscow, Nauka Publ., 1973. 751 p. (in Russian)
16. Kheifetc A.L. Algoritmy modelirovanija konik v pakete AutoCAD [Algorithms modeling in AutoCAD package Conic]. Sovershenstvovanie podgotovki uchashhihsja i studentov v oblasti grafiki, konstruirovanija i standartizacii. Mezhvuzovskij nauchno-metodicheskij sbornik [Perfection of preparation of pupils and students in the field of graphics, design and standardization. Interuniversity scientific-methodological collection]. Saratov, SGTU Publ., 2013, pp. 34–39. (in Russian)
17. Kheyfets A.L. Geometricheskaya tochnost kompyuternyih algoritmov konstruktivnyih zadach [Geometrical accuracy of computer algorithms for constructive problems]. Materialyi VI Mezhdunarodnoy nauchno-prakticheskoy internet konferentsii, Perm, fevral-mart 2016 g. [Materials of the VI International Scientific and Practical Internet Conference, Perm, February-March 2016]. 2016, Permskiy natsionalnyiy issledovatelskiy politehnicheskiy universitet Publ., V. 3, pp. 367–387. (in Russian). Available at: http://elibrary.ru/ item.asp?id=27220185; http://dgng.pstu.ru/conf2016/papers/ 74/
18. Kheifetc A.L. Nachertatelnaya geometriya kak “beg v meshkah” [Descriptive geometry as a factor limiting the development of geometric modeling]. Problemyi kachestva graficheskoy podgotovki studentov v tehnicheskom vuze. Materialyi V Mezhdunarodnoy nauchno-prakticheskoy internetkonferentsii. KGP-2015 ["Problems of quality graphic preparation of students in a technical college. Proceedings of the V International scientific and practical Internet-conference. MSE 2015 "]. Perm: PGTU Publ., 2015, pp. 292–325. (in Russian). Available at: http://dgng.pstu.ru/conf2015/papers/ 72/ http://dgng.pstu.ru/media/files/%D0%A1%D0% B1%D0%BE%D1%80%D0%BD%D0%B8%D0%BA_% D0%9A%D0%93%D0%9F-2015.pdf/
19. Kheifetc A.L., Vasil′eva V.N. Parametrizatsiya kak metod postroeniya sfer Dandelena dlya proizvolnyih kvadrik vrascheniya [Parameterization as a method of constructing spheres of Dandaluna for arbitrary rotation quadrics]. IV Mezhdunarodnaya internet-konferentsiya «Problemyi kachestva graficheskoy podgotovki studentov v tehnicheskom vuze: traditsii i innovatsii» KGP-2014 [IV International Internet Conference "Problems of quality graphic preparation of students in a technical college: tradition and innovation" MSE 2014]. Available at: http://dgng.pstu.ru/conf2014/papers/ 98/
20. Kheyfets A., Vasil′eva V. Realizatsiya obobshchennoy teoremy Dandelena dlya proizvolnykh kvadrik vrashcheniya v AutoCAD [Generalized Dandelin’s Theorem Implementation for Arbitrary Rotation Quadrics in AutoCAD]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 2, pp. 9–14. (in Russian). DOI: 10.12737/5584.
21. Kheifetc A.L. Sravnenie metodov nachertatel'noj geometrii i 3D komp'juternogo geometricheskogo modelirovanija po tochnosti, slozhnosti i jeffektivnosti [Comparison of the methods of descriptive geometry and 3D computer geometric modeling for accuracy, complexity and effectiveness]. Vestnik Juzhno-Ural'skogo gosudarstvennogo universiteta. Serija “Stroitel'stvo i arhitektura” [Bulletin of the South Ural State University. Series “Construction Engineering and Architecture”]. Cheljabinsk: JuUrGU Publ., 2015, V. 15, I. 4, pp. 49–63. (in Russian)
22. Kheifetc (Kheyfets) A. L. 3D-modeli i algoritmyi kompyuternoy parametrizatsii pri reshenii zadach konstruktivnoy geometrii (na nekotoryih istoricheskih primerah) [3d Models and Algorithms for computer-based parameterization for the decision of tasks of constructive Geometry (at some historical Examples)]. Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Seriya “Kompyuternyie tehnologii, upravlenie, radioelektronika” [Bulletin of the South Ural State University. Series "Computer technologies, automatic control & radioelectronics"]. Cheljabinsk: JuUrGU Publ., 2016, V. 16, I. 2, pp. 24–42. (in Russian). DOI: http:// dx.doi.org/10.14529/ctcr160203. Available at: https://vestnik. susu.ru/ctcr/article/view/4909/
23. Chetveruhin N.F., Levitskiy V.S., Pryanishnikova Z.I., Tevlin A.M., Fedotov G.I. Nachertatelnaya geometriya [Descriptive geometry]. Moscow, Vyissh. shk. Publ., 1963. 420 p. (in Russian)
24. Chetveruhin N.F. Proektivnaya geometriya [Projective geometry]. Moscow, Uchpedgiz Publ., 1961. 360 p. (in Russian)
25. SHal' M. Istoricheskij obzor proiskhozhdeniya i razvitiya geometricheskih metodov [Historical review of the origin and development of geometric methods]. O sposobe postroeniya fokusov i dokazatel'stva ikh svoystv na kosom konuse [On the method of constructing foci and proving their properties on an oblique cone]. Moscow, Mosk. mat. o-vo Publ., 1883. 748 p. (in Russian)
26. Dandelin G. Mémoire sur l’hyperboloïde de révolution, et sur les hexagones de Pascal et de M. Brianchon // Nouveaux mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles, V. III., 1826 (pp. 3–16). Hyperboloids of revolution and the hexagons of Pascal and Brianchon. English translation. URL: http://www.math.ubc.ca/~cass/dandelin. pdf/
27. Apostol T., Mnatsakanian M. New descriptions of conics via twisted cylinders, focal disks, and directors. Amer. math. monthly, 115(9): 795–812, 2008. Available at: http://www. mamikon.com/USArticles/NewConics.pdf/
28. Nilov F.K. A generalization of the Dandelin theorem. / Journal of Classical Geometry, 2013, Volume 2, pp. 57–65. Available at: http://jcgeometry.org/Articles/Volume2/JCG- 2013V2pp57-65.pdf/
