ENGINEERING GEOMETRY AND COMPUTER GRAPHICS

Automatic receipt of projective geometry drawings of Johnson bodies

Vestnik MGSU 6/2014
  • Ivashchenko Andrey Viktorovich - Union of Moscow Architects 90/17 Shosseynaya str., Moscow, 109383, Russian Federation; ivashchenkoa@inbox.ru, Union of Moscow Architects, 7 Granatnyy per., Moscow, 123001, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Kondrat’eva Tat’yana Mikhaylovna - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Technical Sciences, Associate Professor, chair, Department of Descriptive Geometry and Graphics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 179-183

The article analyzes the possibilities of polyhedral structures’ formation basing on the automated construction of orthographic drawings (trace diagrams) derived from Johnson bodies. Projective Graphical method makes it possible to simulate the new multi-faceted forms with the help of the trace diagrams selected as a basis of a polyhedron. The computer program developed for this aim allows receiving both trace diagrams and diverse visual images of newly created polygonal shapes. Due to the large number of possible solutions it is proposed to use trace diagrams themselves (based on their degree of symmetry) as a tool to assess the feasibility of using this or that Johnson body as a basis for further shaping.

DOI: 10.22227/1997-0935.2014.6.179-183

References
  1. Johnson N.W. Convex Polyhedra with Regular Faces. Can. J. Math. 1966, vol. 18, no. 1, pp. 169—200. DOI: http://dx.doi.org/10.4153/CJM-1966-021-8.
  2. Gurin A.M. K istorii izucheniya vypuklykh mnogogrannikov s pravil'nymi granyami. Sibirskie elektronnye matematicheskie izvestiya [On the Studying History of Convex Polyhedra with Regular Faces]. 2010, no. 7, pp. A.5—A.23. Available at: http://semr.math.nsc.ru/v7/a5-23.pdf. Date of access: 29.11.13
  3. Wenninger M. Polyhedron Models. Cambridge University Press, 1974.
  4. Dutch Steven. Polyhedra with Regular Polygon Faces. Johnson Polyhedra. Available at: http://www.uwgb.edu/dutchs/symmetry/johnsonp.htm. Date of access: 18.01.2014.
  5. Zalgaller V.A. Vypuklye mnogogranniki s pravil'nymi granyami [Convex Polyhedra with Regular faces]. Records of Scientific Workshop. LOMI, 2, Nauka Publ., Moscow-Leningrad, 1967.
  6. Sutton Daud. Platonic & Archimedean Solids. The Geometry of Space. NY, Walker & Company, 2002, 64 p.
  7. Ivashchenko A.V., Kondrat'eva T.M. Proektivograficheskie chertezhi mnogokomponentnykh sistem mnogogrannikov [Shape Generation by Means of a New Method of Orthographic Representation (“Proektivografiya”): Drawings of Multi-Component Polyhedra]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 6, pp. 155—160.
  8. Ivashchenko A.V., Kondrat'eva T.M. Proektivograficheskiy analiz mnogogrannikov Dzhonsona [Analysis of Johnson’s Polyhedra Using Projective Geometry Techniques]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 5, pp. 226—229.
  9. Gamayunov V.N. Proektivografiya [Projective Geometry]. Moscow, MGPI Publ., 1976, 25 p.
  10. Kalinicheva M.M., Zherdyaev E.V., Novikov A.I. Nauchnaya shkola ergodizayna, VNIITE: predposylki, istoki, tendentsiya stanovleniya. Monografiya. [Scientific School of Energy Design, All-Russian Research Institute of Technical Aesthetics: Background, Origins, Establishment Tendency]. Moscow, VNIITE Publ., Orenburg, IPK GOU OGU Publ., 2009, 368 p.
  11. Sobolev N.A. Obshchaya teoriya izobrazheniy [General Theory of Image] Moscow, Arkhitektura-S Publ., 2004, pp. 489—491.
  12. Ivashchenko A.V. Modeli predstavleniya elementov sistemy proektivograficheskikh epyur i algoritm ikh opredeleniya [Representation Models of the System Elements of Project Geometry Diagrams and their Definition Algorithm]. Molodye golosa: sbornik nauchnoissledovatel’skikh rabot aspirantov i soiskateley [Young Voices: Collection of Scientific Works of Postgraduate Students and Doctoral Candidates]. Moscow, MGOPU Publ., 2000, no. 2.
  13. Nikulin E.A. Komp'yuternaya geometriya i algoritmy mashinnoy grafiki. [Geometry and Algorithms for Computer Graphics]. Saint Petersburg, BKhV-Peterburg Publ., 2003.
  14. Korn G., Korn T. Spravochnik po matematike [Handbook of Mathematics]. Moscow, Nauka Publ., 1970.
  15. Gamayunov V.N., Filin Yu.N. Proektivografiya konfiguratsii Dezarga [Projective Geometry of Desargues Configuration]. Formoobrazovanie v stroitel'stve i arkhitekture: sbornik nauchnykh trudov MISI [Shaping in Construction and Architecture: Collection of Scientific Works of Moscow Institute of Construction and Engineering]. Moscow, MISI Publ., 1986, Part I «Proektivografiya» [Projective Geometry], pp. 105—109.
  16. Gol'tseva R.I. Geometriya mnogogrannykh n-epyurnykh sistem [Polyhedral Geometry of n-Curve Systems]. Formoobrazovanie v stroitel'stve i arkhitekture: sbornik nauchnykh trudov [Shaping in Construction and Architecture: Collection of Scientific Works]. Moscow, MISI Publ., 1986, pp. 175—223.

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Projective configurations in projectivegeometrical drawings

Vestnik MGSU 5/2015
  • Ivashchenko Andrey Viktorovich - Union of Moscow Architects 90/17 Shosseynaya str., Moscow, 109383, Russian Federation; ivashchenkoa@inbox.ru, Union of Moscow Architects, 7 Granatnyy per., Moscow, 123001, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Kondrat’eva Tat’yana Mikhaylovna - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Technical Sciences, Associate Professor, chair, Department of Descriptive Geometry and Graphics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 141-147

The article focuses on the optimization of the earlier discussed computer method of obtaining new forms of polyhedra based on projective geometry drawings (trace Diagrams).While working on getting new multifaceted forms by projective geometry methods based on the well-known models of polyhedra on the first stage of the work it is required to calculate the parameters of projective geometry drawings, and then to build them. This is an often used apparatus of analytical geometry. According to it, at first the parameters of the polyhedron (core system of planes) are calculated, then we obtain the equation of the plane of the face of the polyhedron, and finally we obtain the equations of lines the next plane faces on the selected curve plane. At each stage of application such a method requires the use of the algorithms of floating point arithmetic, on the one hand, leads to some loss of accuracy of the results and, on the other hand, the large amount of computer time to perform these operations in comparison with integer arithmetic operations.The proposed method is based on the laws existing between the lines that make up the drawing - the known configurations of projective geometry (complete quadrilaterals, configuration of Desargues, Pappus et al.).The authors discussed in detail the analysis procedure of projective geometry drawing and the presence of full quadrilaterals, Desargues and Pappus configurations in it.Since the composition of these configurations is invariant with respect to projective change of the original nucleus, knowing them, you can avoid the calculations when solving the equations for finding direct projective geometry drawing analytically, getting them on the basis of belonging to a particular configuration. So you can get a definite advantage in accuracy of the results, and in the cost of computer time. Finding these basic configurations significantly enriches the set of methods and the use of projective geometry drawings.

DOI: 10.22227/1997-0935.2015.5.141-147

References
  1. Gamayunov V.N. Proektivografiya. Geometricheskie osnovy khudozhestvennogo konstruirovaniya dlya aspirantov slushateley FPK i studentov khuzhozhestvenno-graficheskogo fakul’teta [Projectography. Geometric Foundations of Artistic Design for Postgraduate Students of FPK and Students of Artistic-Graphical Department]. Moscow, MGPI Publ., 1976, 25 p. (In Russian)
  2. Gol’tseva R.I. Geometriya mnogogrannykh n-epyurnykh sistem [Polyhedral Geometry of n-Curve Systems]. Formoobrazovanie v stroitel’stve i arkhitekture: sbornik nauchnykh trudov [Shaping in Construction and Architecture: Collection of Scientific Works]. Moscow, MISI Publ., 1986, pp. 175—223. (In Russian)
  3. Sobolev N.A. Obshchaya teoriya izobrazheniy [General Theory of Image] Moscow, Arkhitektura-S Publ., 2004, pp. 489—491. (In Russian)
  4. Kalinicheva M.M., Zherdyaev E.V., Novikov A.I. Nauchnaya shkola ergodizayna VNIITE: predposylki, istoki, tendentsiya stanovleniya : monografiya [Scientific School of Ergodesign All-Russian Research Institute of Technical Aesthetics: Prerequisites, Origins, Generation Tendency : Monograph]. Moscow, VNIITE Publ., Orenburg, IPK GOU OGU Publ., 2009, 368 p. (In Russian)
  5. Vennidzher M. Modeli mnogogrannikov [Models of Polyhedra]. Moscow, Mir Publ.,1974, 236 p. (In Russian)
  6. Zalgaller V.A. Vypuklye mnogogranniki s pravil’nymi granyami [Convex Polyhedra with Regular Faces]. Zapiski nauchnykh seminarov LOMI [Records of Scientific Workshops of LOMI]. Moscow-Leningrad, Nauka Publ., 1967, vol. 2, pp. 5—221. (In Russian)
  7. Dutch S. Polihedra with Regular Polygon Faces. Available at: http://www.uwgb.edu/DUTCHS/symmetry/johnsonp.htm. Date of access: 18.11.2014.
  8. Sutton D. Platonic & Archimedean Solids: the Geometry of Space/written and Illustrated. New York, Walker & Company, 2002, 64 p.
  9. Gurin A.M. K istorii izucheniya vypuklykh mnogogrannikov s pravil’nymi granyami [Background of Study of Convex Polyhedra with Regular Faces]. Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic News of Mathematics]. 2010, vol. 7, pp. 5—23. (In Russian)
  10. Alsina C. Mir matematiki : v 40 tomakh. Tom 23. Tysyacha graney geometricheskoy krasoty. Mnogogranniki [The World of Mathematics : in 40 Volumes. Vol. 23. Thousand Faces of Geometrical Beauty. Polyhedrons]. Translated from Spanish]. Moscow, De Agostini Publ., 2014, 144 p. (In Russian)
  11. Ivashchenko A.V. Modeli predstavleniya elementov sistemy proektivograficheskikh epyur i algoritm ikh opredeleniya [Representation Models of the System Elements of Project Geometry Diagrams and their Definition Algorithm]. Molodye golosa: sbornik nauchno-issledovatel’skikh rabot aspirantov i soiskateley [Young Voices: Collection of Scientific Works of Postgraduate Students and Doctoral Candidates]. Moscow, MGOPU Publ., 2000, no. 2, pp. 12—19. (In Russian)
  12. Ivashchenko A.V., Kondrat’eva T.M. Proektivograficheskie chertezhi mnogokomponentnykh sistem mnogogrannikov [Shape Generation by Means of a New Method of Orthographic Representation (“Proektivografiya”): Drawings of Multi-Component Polyhedra]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2012, no. 6, pp. 155—160. (In Russian)
  13. Ivashchenko A.V., Kondrat’eva T.M. Proektivograficheskiy analiz mnogogrannikov Dzhonsona [Analysis of Johnson’s Polyhedra Using Projective Geometry Techniques]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 5, pp. 226—229. (In Russian)
  14. Ivashchenko A.V., Kondrat’eva T.M. Avtomatizatsiya polucheniya proektivograficheskikh chertezhey tel Dzhonsona [Automatic Receipt of Projective Geometry Drawings of Johnson Bodies]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2014, no. 6, pp. 179—183. (In Russian)
  15. Ivashchenko A.V., Znamenskaya E.P. Konfiguratsiya Dezarga v arkhitekturnom i dizayn-proektirovanii [Configuration of Desargue in Architectural and Design Engineering]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2014, no. 9, pp. 154—160. (In Russian)
  16. Nikulin E.A. Komp’yuternaya geometriya i algoritmy mashinnoy grafiki [Geometry and Algorithms for Computer Graphics]. Saint Petersburg, BKhV-Peterburg Publ., 2003, 560 p. (In Russian)
  17. Chetverukhin N.F. Vysshaya geometriya [Higher Geometry]. Moscow, Uchpedgiz Publ., 1939, 144 p. (In Russian)
  18. Young J.W., Veblen O. Projective Geometry. University of Michigan, 1910, 360 p.
  19. Hartshorne R. Foundations of Projective Geometry. Ishi Press, 2009, 190 p.
  20. Filin Yu.N., Veselov V.I., Georgievskiy O.V. Innovatsionnoe preobrazovanie formografiki kubicheskikh modeley v svete resheniya problem razvitiya ekologicheski znachimykh form [Innovative Transformation of Form Graphics of Cubic Models in Frames of Solving the Problems of Ecologically Essential Forms Development]. Innovatsii: perspektivy, problemy, dostizheniya : sbornik trudov Mezhdunarodnoy nauchno-prakticheskoy konferentsii (Moskva 27 maya 2013 g.) [Innovations: Prospects, Problems, Achievements : Collection of Works of International Science and Practice Conference (Moscow, May 27, 2013)]. Moscow, REU im. G.V. Plekhanova Publ., 2013, pp. 277— 282. (In Russian)
  21. Kartavtsev I.S., Veselov V.I., Georgievskiy O.V., Filin Yu.N. Arkhikub-izokonstruktor transformatsii formografiki [ArchicubeIsoconstructor of Form Graphics Transformation]. Ekonomicheski effektivnye i ekologicheski chistye innovatsionnye tekhnologii : sbornik trudov Mezhdunarodnoy nauchno-prakticheskoy konferentsii [Economically Efficient and Environmentally Friendly Innovative Technologies : Collection of Works of International Science and Practice Conference]. Moscow, REU im. G.V. Plekhanova Publ., 2013, pp. 139—143. (In Russian)

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