Chelyabinsk, Chelyabinsk, Russian Federation
Birational (Cremona) correspondences of two planes П, П' or Cremona transformations on the combined plane П = П' represent an effective tool for design of smooth dynamic curves and two-dimensional lines. The simplest birational correspondence is a quadratic map Ω of plane fields on one another. In the projective definition of the quadratic correspondence can participate two pairs of imaginary complex conjugate F-points set as double points of elliptic involutions on the lines associated with the third pair of F-points. In this case, the imaginary projective F-bundles cannot be used for generation of points corresponding in Ω. A generic constructive algorithm for design of corresponding points in a quadratic mapping Ω(П ↔ П'), set both by real and imaginary F-points is proposed in this paper. The algorithm is based on the use of auxiliary projective correspondence Δ between the points of the planes П, П' and Hirst’s transformation Ψ with the center in the one of real F-points. A theorem on the existence of an invariant conic common to Ω and Δ mappings has been proved. Has been demonstrated a possibility for quadratic mapping’s presentation as a product of collinearity and Hirst’s transformation: Ω = ΔΨ. Has been considered a solution for auxiliary problems arising during the generic constructive algorithm’s implementation: buildup a conic section, that is incident to imaginary points, and plotting the corresponding points in collinearity set by imaginary points. It has been demonstrated that there are two or four possible versions of collinearity for plane fields П, П', set by with participation of the imaginary corresponding points, due to an uncertainty related to the order of their relative correspondence. Have been completely solved the problem of mapping a straight line in a conic section within the quadratic Cremona correspondance of fields П ≠ П', set by a pair of real fundamental points, and two pairs of imaginary ones. It has been demonstrated that in general case the problem has two solutions. The obtained results are useful for the development of the geometric theory related to imaginary elements and this theory’s application in linear and non-linear descriptive geometry, operating projective images of first and second orders.
birational quadratic correspondance, imaginary fundamental points, elliptic involution, inversion, Girst’s transformation.
Введение
В полях П, П′ указаны две пары проективных пучков
F1 ( j3, j2,a1 ) F1 ( j2 , j3 ,a1 ), (1)
F j j a F j j a 2 3 1 2 2 1 3 2 ( , , ) ( , , ). (2)
Выражения (1, 2) устанавливают точечное соответствие Ω плоских полей П и П′ (почти всюду взаимно однозначное), при котором точке A = a1 ∩ a2 поля П соответствует точка A = a a 1 2 поля П′, и наоборот, точке A′ поля П′ отвечает точка A поля П (рис. 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. Kompleksnaya geometriya - evklidova i psevdoevklidova [Complex geometry - Euclidean and pseudo-Euclidean]. Moscow, IPTs «Maska» Publ., 2013. 216 p.
6. 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.
7. Girsh A.G. Naglyadnaya mnimaya geometriya [Transparent imaginary geometry]. Moscow, IPTs «Maska» Publ., 2008. 216 p.
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. Entsiklopediya elementarnoj matematiki [Encyclopedia of elementary mathematics], Moscow, Fizmatlit, 1963, v. 4, 567 p.
