Chelyabinsk, Chelyabinsk, Russian Federation
Kassel', Germany
Second-order curves are used as shape-generating elements in the design of technical devices and architectural structures. In such a case, a need for reconstruction task solution may emerge. The reconstruction is called the definition of the main axes and asymptotes of the second-order curve by its incomplete image containing n points and m tangents (n + m = 5). In CAD graphical systems there is no possibility for construction of the second order curve, given by real and imaginary points and tangents. Therefore, the second-order curve reconstruction cannot be made with the standard set of computer graphics tools. In this paper are proposed geometrically accurate algorithms for reconstruction of the secondorder curve, given by a mixed set of real and imaginary elements. A specialized software package has been developed for constructive realization of these algorithms. Imaginary geometric images are pair-conjugated, so there are only seven possible combinations of given data with imaginary elements participation: five points, two of which are imaginary ones; five points, four of which are imaginary ones; three real points, two imaginary tangents; a real point, four imaginary tangents; a real point, two imaginary points, two imaginary tangents; a real point, two imaginary points, two real tangents; two real points, two imaginary points, a real tangent. For reconstruction problem solution is used the main property of polar matching: if P and p are the pole and polar relative to the conic g, the harmonic homology with center P and axis p transforms the curve g in itself. The method of solution based on projective transformation of required conic into a circle. It has been shown that in some cases for reconstruction problem solution its necessary to apply the quadratic involution conversion, resting on plane by a conic beam. The developed technique and software package expand the capabilities of the computer geometric simulation for processes occurring with the second-order curves participation.
harmonic homology, elliptic involution, polarity, conic beam, autopolar triangle, quadratic involution
Кривые второго порядка (КВП, конические сечения, коники) применяются как для изображения различных кинематических процессов (траектории, орбиты и т.п.), так и в качестве формообразующих элементов при проектировании технических устройств и архитектурных сооружений. При этом может возникать потребность в решении задачи реконструкции. Реконструкцией КВП называют определение метрики (главных осей, асимптот) кривой второго порядка по ее неполному изображению, содержащему n точек и m касательных (n + m = 5).
Известны графические алгоритмы определения метрики КВП для случая, когда заданные элементы (точки и касательные) действительны [4; 17; 18]. Для случая, когда некоторые точки и касательные к искомой КВП — мнимые (комплексно сопряженные), алгоритмы определения метрики КВП отсутствуют.
Цель работы — составить геометрически точный алгоритм реконструкции кривой второго порядка, заданной смешанным набором действительных и мнимых элементов (точек и касательных). Геометрически точным алгоритмом называют последовательность построений с использованием только двух графических примитивов — прямой линии и окружности [1].
1. Volkov V.Ya. Elementy matematizatsii teoreticheskikh osnov nachertatel´noy geometrii [elements mathematization of the theoretical foundations of descriptive geometry]. Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 1, pp. 3-15. DOI:https://doi.org/10.12737/10453.
2. Voloshinov D.V. Konstruktivnoe geometricheskoe modelirovanie. Teoriya, praktika, avtomatizatsiya [Constructive geometrical modeling. Theory, practice, automation]. Saarbrucken: Lambert Academic Publ., 2010. 355 p.
3. Vol´berg O.A. Osnovnye idei proektivnoy geometrii [Basic ideas of projective geometry]. Moscow, Uchpedgiz Publ., 1949. 188 p.
4. Geronimus Ya.L. Geometricheskiy apparat teorii sinteza ploskikh mekhanizmov [Geometric theory of machine synthesis of planar mechanisms]. Moscow, Fiziko-matematicheskaya literature Publ., 1962. 399 p.
5. Girsh A.G. Naglyadnaya mnimaya geometriya [Transparent imaginary geometry]. Moscow, IPTs «Maska» Publ., 2008. 216 p.
6. Girsh A.G. Kompleksnaya geometriya - evklidova i psevdoevklidova [Complex geometry - Euclidean and pseudo-Euclidean]. Moscow, IPTs «Maska» Publ., 2013. 216 p.
7. Girsh A.G. Mnimosti v geometrii [Imaginaries in Geometry]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 2, pp. 3-8. DOI:https://doi.org/10.12737/5583.
8. Girsh A.G. Fokusy algebraicheskikh krivykh [Focuses algebraic curves]. Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 3, pp. 4-17. DOI:https://doi.org/10.12737/14415.
9. Glagolev N.A. Proektivnaya geometriya [Projective geometry]. Moscow, Vysshaya shkola Publ., 1963. 343 p.
10. Golovanov N.N. Geometricheskoe modelirovanie [Geometric modeling]. Moscow, Fiziko-matematicheskaya literatura Publ., 2012. 472 p.
11. Ivanov G.S. Konstruirovanie tekhnicheskikh poverkhnostey. Matematicheskoe modelirovanie na osnove nelineynykh preobrazovaniy [Construction of technical surfaces. Mathematical modeling based on nonlinear transformations]. Moscow, Mashinostroenie Publ., 1987. 192 p.
12. Ivanov G.S. Konstruktivnyy sposob issledovaniya cvoystv parametricheski zadannykh krivykh [A constructive way to explore the Properties of parametrically defined curves]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 3, pp. 3-6. DOI:https://doi.org/10.12737/6518.
13. Ivanov G.S. O zadachakh nachertatel´noy geometrii s mnimymi resheniyami [On the tasks of descriptive geometry with imaginary solutions]. Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 2, pp. 3-8. DOI:https://doi.org/10.12737/12163.
14. Kleyn F. Vysshaya geometriya [Higher geometry]. Moscow, URSS Publ., 2004. 400 p.
15. Kleyn F. Elementarnaya matematika s tochki zreniya vysshey [Elementary mathematics from the point of view of the highest]. Moscow, Nauka Publ., 1987, V. 2, 416 p.
16. Kokster Kh.S.M. Deystvitel´naya proektivnaya ploskost´ [The real projective plane]. Moscow, Fizmatlit Publ., 1959. 280 p.
17. Korotkiy V.A. Sinteticheskie algoritmy postroeniya krivoy vtorogo poryadka [Synthetic algorithms for constructing a second-order curve]. Vestnik komp´yuternykh i informatsionnykh tekhnologiy [Herald of computer and information technologies]. 2014, I. 11, pp. 20-24. DOI:https://doi.org/10.14489/issn.1810-7206.
18. Korotkiy V.A. Proektivnoe postroenie koniki [The projected construction of conic]. Chelyabinsk, YuUrGU Publ., 2010. 94 p.
19. Korotkiy V.A. Kvadratichnoe preobrazovanie ploskosti, ustanovlennoe puchkom konicheskikh secheniy [Quadratic transformation of the plane, set beam conic sections]. Omskiy nauchnyy vestnik. Seriya «Pribory, mashiny i tekhnologii» [Omsk Scientific Bulletin. Series "devices, machines and technology"]. 2013, I. 1, pp. 9-14.
20. Peklich V.A. Vysshaya nachertatel´naya geometriya [Higher descriptive geometry]. Moscow, ASV Publ., 2000. 344 p.
21. Peklich V.A. Mnimaya nachertatel´naya geometriya [Imaginary descriptive geometry]. Moscow, ASV Publ., 2007. 104 p.
22. Programma dlya EVM «Postroenie krivoy vtorogo poryadka, prokhodyashchey cherez dannye tochki i kasayushcheysya dannykh pryamykh» [The computer program "Building a second order curve through the data points and data relating to direct"]. Svidetel´stvo o gosudarstvennoy registratsii № 2011611961 ot 04.03.2011 g. [Short Certificate of state registration number 2011611961 from 04.03.2011].
23. Savel´ev Yu.A. Grafika mnimykh chisel [Graphic imaginary numbers]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 1, pp. 22-23. DOI:https://doi.org/10.12737/465.
24. Sal´kov N.A. Nachertatel´naya geometriya - baza dlya geometrii analiticheskoy [Descriptive Geometry - the base for geometry analysis]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 1, pp. 44-54. DOI:https://doi.org/10.12737/18057.
25. Seregin V.I. Mezhdistsiplinarnye svyazi nachertatel´noy geometrii i smezhnykh razdelov vysshey matematiki [Interdisciplinary communication descriptive geometry and adjacent sections of higher mathematics]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 3/4, pp. 8-12. DOI:https://doi.org/10.12737/2124.
26. Suvorov F.M. Ob izobrazhenii voobrazhaemykh tochek i voobrazhaemykh pryamykh na ploskosti i o postroenii krivykh liniy vtoroy stepeni, opredelyaemykh s pomoshch´yu voobrazhaemykh tochek i kasatel´nykh [On image imaginary imaginary points and lines in the plane and on the construction of curves of the second degree, as determined by the imaginary points and tangents]. Kazan´, Tipografiya imperatorskogo Universiteta Publ., 1884. 130 p.
27. Foks A. Vychislitel´naya geometriya. Primenenie v proektirovanii i na proizvodstve [Computational geometry. The application in the design and production]. Moscow, Mir Publ., 1982. 304 p.
28. Shikin E.V. Krivye i poverkhnosti na ekrane komp´yutera [The curves and the surface on the computer screen]. Dialog-MIFI Publ., 1996. 240 p.
