employee
Russian Federation
A method for obtaining of polyhedral structures when modeling polyhedra using the projective graphic method by changing parameters of a convex polyhedron, taken as a kernel, is proposed in this paper. A process for obtaining a kernel of a tetrahedral symmetry group (tetrahedral 24-hedron) has been considered in detail. The kernel parameters influence on the shape change of corresponding polyhedral structures is analyzed. For this, the parameters V, G, and R are used, which are the lengths of radius vectors drawn from the center of the sphere circumscribed around the initial tetrahedron to the vertices, midpoints of the edges, and the centers of this tetrahedron’s facets. It has been shown that the influence quantity of the kernel shape on the shape of the three-dimensional object’s generated elements depends on the degree of proximity of this generated element to the kernel itself; moreover, this influence is such that the near regions depend on the kernel shape to a lesser extent than the distant ones. As examples, shape-forming solutions based on a tetrahedral 24-hedron have been obtained, which reflect a change in the central regions’ shape, and solutions reflecting a change in the shape of regions distant from the center. In addition, shape-forming solutions based on dihedral symmetry have been also considered. Varying the kernel parameters within certain limits by a developed computer program can serve as a tool for designing such forms of objects that are difficult to formalize in the traditional way. A visual analysis of projective graphic drawings for objects deformed in a certain way in order to obtain acceptable results on three-dimensional objects is carried out. In conclusion, a comparison of the proposed shape-forming method with alternative options for obtaining similar results without using projectography is given. The proposed technique can be used by architects and designers while constructing forms with predetermined properties.
shaping of polyhedral structures, projective-graphic method, kernels of tetrahedral and dihedral symmetry, influence of kernel parameters, variation of kernel parameters as designer’s tool
1. Zolotye sechenie i zolotye pryamougol'niki pri postroenii ikosaedra, dodekaedra i tel Arhimeda, osnovannyh na nih [Golden section and golden rectangles in the construction of the icosahedron, dodecahedron and the bodies of Archimedes based on them]. Geometriya i grafika [Geometry and graphics]. 2019, V. 7, I. 2, pp. 47–55. DOI: 10.12737/article_5d2c1ceb9f91b1.21353054. (in Russian)
2. Gamayunov V.N. Proektivografiya [Projectivography]. Moscow: MGPI Publ., 1976. 86 p. (in Russian)
3. Gol'ceva R.I. Geometriya mnogogrannyh n-epyurnyh sistem [Geometry of polyhedral n-diagram systems]. Formoobrazovanie v stroitel'stve i arhitekture [Forming in construction and architecture]. Moscow: MISI im. Kujbysheva Publ., 1986, pp. 175–223. (in Russian)
4. Zhiharev L.A. Fraktaly v trekhmernom prostranstve. I-fraktaly [Fractals in three-dimensional space. I-fractals]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 3, pp. 51–66. DOI: 10.12737/article_59bfa55ec01b38.55497926. (in Russian)
5. Zhiharev L.A. Fraktal'nye razmernosti [Fractal dimensions]. Geometriya i grafika [Geometry and Graphics]. 2018, V. 6, I. 3, pp. 33–47. DOI: 10.12737/article_5bc45918192362.77856682. (in Russian)
6. Ivanov V.N. Osnovy razrabotki i vizualizacii ob"ektov analiticheskih poverhnostej i perspektivy ih ispol'zovaniya v arhitekture i stroitel'stve [Fundamentals of the development and visualization of objects of analytical surfaces and the prospects for their use in architecture and construction]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 4, pp. 3–14. DOI: 10.12737/article_5a17f590be3f51.37534061. (in Russian)
7. Ivashchenko A.V. Modeli predstavleniya elementov sistemy proektivograficheskih epyur i algoritm ih opredeleniya [Representation models of elements of a system of projective graphic diagrams and an algorithm for their determination]. Golosa molodyh [Voices of the young]. MGOPU Publ., 2000, I. 2, pp. 12–19. (in Russian)
8. Ivashchenko A.V. Proektivograficheskie chertezhi mnogokomponentnyh sistem mnogogrannikov [Projective design drawings of multicomponent polyhedron systems]. Vestnik MGSU [Bulletin of MGSU]. 2012, I. 6, pp. 155–160. (in Russian)
9. Ivashchenko A.V. Proektivograficheskij analiz mnogogrannikov Dzhonsona [Projectivographic analysis of Johnson's polyhedra]. Vestnik MGSU [Vestnik MGSU]. 2013, I. 5, pp. 226–229. (in Russian)
10. Ivashchenko A.V. Avtomatizaciya polucheniya proektivograficheskih chertezhej tel Dzhonsona [Automation of receiving projective drawings of Johnson's bodies]. Vestnik MGSU [Vestnik MGSU]. 2014, I. 6, pp. 179–183. (in Russian)
11. Ivashchenko A.V. Proektivnye konfiguracii na proektivograficheskih chertezhah [Projective configurations on projective drawings]. Vestnik MGSU [Vestnik MGSU]. 2015, I. 5, pp. 141–147. (in Russian)
12. Ivashchenko A.V. Osobennosti preobrazovaniya sistem koordinat na proektivograficheskih chertezhah [Features of the transformation of coordinate systems in projective drawings]. Nauchnoe obozrenie [Scientific Review]. Moscow, 2016, I. 9, pp. 47–51. (in Russian)
13. Ivashchenko A.V. Ob ispol'zovanii polyarnoj sistemy koordinat v proektivograficheskih chertezhah [On the use of the polar coordinate system in projective drawings]. Vestnik MGSU [Bulletin of MGSU]. 2016, I. 11, pp. 124–131. (in Russian)
14. Ivashchenko A.V. Hudozhestvennoe proektirovanie tekstil'nogo risunka na osnove proektivograficheskih chertezhej tel Dzhonsona [Artistic design of textile patterns based on projective drawings of Johnson's bodies]. Tekhnologiya tekstil'noj promyshlennosti [Technology of the textile industry]. 2017, I. 3 (369), pp. 189–192. (in Russian)
15. Ivashchenko A.V. O metode formoobrazovaniya v arhitekture i dizajne, osnovannom na mnogoyadernyh proektivograficheskih sistemah [About the method of shaping in architecture and design based on multi-core projective systems]. Innovacii i investicii [Innovations and investments]. 2017, I. 8, pp. 132–136. (in Russian)
16. Klejn F. Lekcii ob ikosaedre i reshenii uravneniya 5-j stepeni [Lectures on the icosahedron and the solution of equations of the 5th degree]. Moscow: Nauka Publ., 1989, p. 336. (in Russian)
17. Romanova V.A. Vizualizaciya pravil'nyh mnogogrannikov v processe ih postroeniya [Visualization of regular polyhedrons in the process of their construction]. Geometriya i grafika [Geometry and graphics]. 2019, V. 7, I. 1, pp. 55–67. DOI: 10.12737/article_5c91ffd0916d52.90296375. (in Russian)
18. Yurkov V.Yu. Approksimaciya mnozhestv pryamyh na ploskosti [Approximation of the sets of lines in the plane]. Geometriya i grafika [Geometry and graphics]. 2019, V. 7, I. 3, pp. 60–69. DOI: 10.12737/article_5dce6cf7ae1d70.85408915. (in Russian)
19. Barsky B. Computer graphics and geometric modeling using Beta-splines. [Text] / Springer Verlag, 1988, — p.157.
20. Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Schwarzkopf. Computational Geometry: Algorithms and Applications. [Text] / Springer, 2000. — p.368.
21. Bruckner M. Vielecke und Vielflache. [Text] / Leipzig, Tuebner, 1900, p.227. Chen J. Computational Geometry: Methods and Applications. [Text] / — Texas A&M University, 1996. — p.228.
22. Coxeter H.S.M. The Fifty Nine Icosahedra (with P. Du Val, H. T. Flather, J. F.Petrie) [Text] / Springer-Verlag, 1982, — p.72
23. Coxeter H. S. M. Regular Poyltops. [Text] / Daver Publications, 1973, —p.368.
24. Foley D.J., van Dam A., Feiner S.K., Hughes J.F. Computer graphics. Principles and practice. [Text] / Addison-Wesley, 1991, — p.1175.
25. Farin G. Curves and surfaces for computer aided geometric design. A practical guide. [Text] / — Academicv Press, 1990, — p.444.
26. Goodman J.E., Joseph O'Rourke. Handbook of Discrete and Computational Geometry. [Text] /. — CRC Press LLC, 1997. — p. 956
27. Johnson N.W. Convex polyhedra with regular faces. [Text] /, Can. J.Math., 18: 1 (1966), pp.169 — 200.
28. Langetepe Elmar, Gabriel Zachmann. Geometric Data Structures for Computer Graphics. [Text] / A K Peters, 2006. — p.362. — ISBN 1568812353.
29. Mount David. Computional Geometry. [Text] / — University of Maryland, 2002. — p. 122.
30. Hormoz Pirzadeh. Computational Geometry with the Rotating Calipers. [Text] / — McGill University, 1999. — p. 118.
31. Wenninger M, Polyhedron Models. [Text] / Cambridge University Press, ISBN 978-0-521-09859-5, 1971. — p. 208.
32. Wenninger M. Dual Models. [Text] / Cambridge University Press, ISBN 978-0-521-54325-5, 1983. — p. 172.
33. Dutch S., Polyhedra with Regular Polygon Faces.
34. http://www.uwgb.edu/dutchs/symmetry/johnsonp.htm
