IMAGINARY STRAIGHT LINES IN CARTESIAN COORDINATE SYSTEM Abstract (English):
A geometric model of imaginary conjugate straight lines a~b, allowing symbolic representation of these lines on the real coordinate plane xy is considered. In order to connect the algebraic and geometric representations of imaginary straight lines, it is proposed to use the “mark” formed by orthogonal d1 ⊥ d2 and main g1~g2 directions of the elliptic involution σ in the pencil V. The specification of two pairs of pulling apart each other real straight lines d1~d2, g1~g2 passing through V, uniquely defines the elliptic involution σ in the pencil V, therefore, the V(d1 ⊥ d2, g1~g2) mark completely defines a pair of imaginary double straight lines a~b of elliptic involution σ(V), that allows consider the mark as an “image” of these imaginary straight lines. When using a mark, it is required to establish a one-to-one correspondence between complex coefficients of imaginary double straight lines equations and a graphically given mark. The direct and inverse problems are solved in this paper. The direct one is creation a mark representing imaginary straight lines, given by its own equations. The inverse one is determination of coefficients for the equations of imaginary lines defined by the mark. The essence of the direct and inverse problems consists in establishing a oneto-one correspondence between the equations of imaginary double straight elliptic involutions σ in the pencil V, and a graphically given mark containing the orthogonal and main directions of this involution. To solve both the direct and inverse problems, the Hirsch theorem (A.G. Hirsch) is used, which establishes a one-to-one correspondence between the complex Cartesian coordinates for a pair of imaginary conjugated points and real coordinates of a special “marker” symbolically representing these points. Have been considered examples of solution for geometric problems involving imaginary lines. In particular, has been solved the problem of constructing a circle passing through a given point and touching imaginary lines defined by its mark V(d1 ⊥ d2, g1~g2). Has been proposed a graphical and analytical algorithm for determining the coefficients of equations of imaginary tangents, traced to a conic section from its inner point.

Keywords:
complex Cartesian coordinates, elliptic involution, orthogonal and principal axes of involution; cyclic points; isotropic straight lines

1. Byushgens S.S. Analiticheskaya geometriya. Pervyj koncentr [Analytic geometry]. Moscow-Leningrad, Gosudarstvennoe uchebno-pedagogicheskoe izdatel'stvo Publ., 1934, 237 p. (in Russian)

2. Vol'berg O.A. Osnovnye idei proektivnoy geometrii [The Basic Ideas of Projective Geometry]. Moscow-Leningrad, Gosudarstvennoe uchebno-pedagogicheskoe izdatel'stvo Publ., 1949. 188 p. (in Russian)

3. Voloshinov D.V. Edinyj konstruktivnyj algoritm postroeniya fokusov krivyh vtorogo poryadka [A Unified Constructive Algorithm for Second-Order Curves’ Foci Creation]. Geometriya i grafika [Geometry and graphics]. 2018, V. 6, I. 2, pp. 47–54. DOI: 10.12737/2308-4898. (in Russian)

4. Hilbert D., Cohn-Vossen S. Naglyadnaya geometriya [Visual geometry]. Moscow, Nauka Publ., 1981. 344 p. (in Russian)

5. Hirsh A.G. Kompleksnaya geometriya – evklidova i psevdoevklidova [Complex geometry - Euclidean and pseudo-Euclidean]. Moscow, LLC "Mask" Publ., 2013. 216 p. (in Russian)

6. Hirsh A.G. Naglyadnaya mnimaya geometriya [Visual imaginary geometry]. Moscow, LLC "Mask" Publ., 2008. 216 p. (in Russian)

7. Hirsh A.G. Nachala kompleksnoj geometrii. Izbrannye zadachi kompleksnoj geometrii s resheniyami. CHast' II ─ 3D [The beginnings of complex geometry. Selected problems of complex geometry with solutions. Part II ─ 3D]. Kassel, Germany. 2014. 112 p. (in Russian)

8. Hirsh A.G. Nachala kompleksnoj geometrii. Sbornik zadach po kompleksnoj geometrii s resheniyami. CHast' I ─ 2D [The beginnings of complex geometry. Collection of problems on complex geometry with solutions. Part I - 2D]. Kassel, Germany. 2012. 191 p. (in Russian)

9. Hirsh A.G., Korotkiy V.A. Graficheskie algoritmy rekonstrukcii krivoj vtorogo poryadka, zadannoj mnimymi elementami [Graphic Reconstruction Algorithms of the Second-Order Curve, given by the Imaginary Elements]. Geometriya i grafika [Geometry and graphics]. 2016, V. 4, I. 4, pp. 19–30. DOI: 10.12737/22840. (in Russian)

10. Hirsh A.G., Korotkiy V.A. Mnimye tochki v dekartovoj sisteme koordinat [Imaginary points in a Cartesian coordinate system]. Geometriya i grafika [Geometry and graphics]. 2019, V. 7, I. 3, pp. 28–35. DOI: 10.12737/article_5dce651d80b827.49830821. (in Russian)

11. Glagolev N.A. Proektivnaya geometriya [Projective Geometry]. Moscow, Vysshaya Shkola Publ., 1963. 344 p. (in Russian)

12. Grafskij O.A., Ponomarchuk YU.V., Holodilov A.A. Geometriya elektrostaticheskih polej [Electrostatic field geometry]. Geometriya i grafika [Geometry and graphics]. 2018, V. 6, I. 1, pp. 10–19. DOI: 10.12737/article_5ad085a6d75bb5.99078854. (in Russian)

13. Klejn F. Vysshaya geometriya [Higher Geometry], Moscow, URSS Publ., 2004. 400 p.

14. Klejn F. Elementarnaya matematika s tochki zreniya vysshej [Elementary Mathematics from the Point of View of Higher]. Moscow, Nauka Publ., 1987. 416 p. (in Russian)

15. Korotkiy V.A. Gomologiya dvukh konicheskikh secheniy [Homology of two conic sections]. Sovershenstvovanie podgotovki uchashchihsya i studentov v oblasti grafiki, konstruirovaniya i standartizacii [Improving the training of students in the field of graphics, design and standardization]. Saratov: SGTU Publ., 2012, pp. 27–33. (in Russian)

16. Korotkiy V.A. Graficheskie algoritmy postroeniya kvadriki, zadannoj devyat'yu tochkami [Graphic algorithms for constructing a quadric defined by nine points]. Geometriya i grafika [Geometry and graphics]. 2019, V. 7, I. 2, pp. 3–12. DOI: 10.12737/article_5d2c1502670779.58031440 (in Russian)

17. Korotkiy V.A. Kvadratichnoe kremonovo sootvetstvie ploskih polej, zadannoe mnimymi F-tochkami [Quadratic Cremonian correspondence of plane fields defined by imaginary F-points]. Geometriya i grafika [Geometry and graphics]. 2017, V. 5, I. 1, pp. 21–31. DOI: 10.12737/ 25120. (in Russian)

18. Korotkiy V.A. Komp'yuternaya vizualizaciya krivoj vtorogo poryadka, prohodyashchej cherez mnimye tochki i kasayushchejsya mnimyh pryamyh [Computer Visualization of a Curve of the Second Order passing through Imaginary Points and touching Imaginary Lines]. Nauchnaya vizualizaciya [Scientific visualization]. 2018, V. 10, I. 1, pp. 56–68. Available at: http://sv-journal.org. (in Russian)

19. Korotkiy V.A., Usmanova E.A. Krivye vtorogo poryadka na ekrane komp'yutera [Second Order Curves on Computer Screen]. Geometriya i grafika [Geometry and graphics]. 2018, V. 6, I. 2, pp. 101–113. DOI: 10.12737/2308-4898. (in Russian)

20. Korotkiy V. A. Preobrazovanie puchka konik v puchok okruzhnostej [Conic beam transformation into a bundle of circles]. Sovershenstvovanie podgotovki uchashchihsya i studentov v oblasti grafiki, konstruirovaniya i standartizacii [Improving the training of students in the field of graphics, design and standardization]. Saratov: SGTU Publ., 2014, pp. 53–57. (in Russian)

21. Kutishchev G.P. Geometriya algebraicheskih uravnenij, razreshimyh v radikalah, s prilozheniyami v chislennyh metodah i vychislitel'noj geometrii [The geometry of algebraic equations solvable in radicals, with applications in numerical methods and computational geometry]. Moscow, Knizhnyj dom "Librokom" Publ., 2012. 168 p. (in Russian)

22. Nemchenko K.E. Analiticheskaya geometriya [Analytic geometry]. Moscow, Eksmo Publ., 2007. 352 p. (in Russian)

23. Peklich V.A. Vysshaja nachertatel'naja geometrija [The Highest Descriptive Geometry]. Moscow, ASV Publ., 2000. 344 p. (in Russian)

24. Peklich V.A. Mnimaya nachertatel'naya geometriya [Imaginary Descriptive Geometry]. Moscow, ASV Publ., 2007. 104 p. (in Russian)

25. Korotkiy V.A. Programma dlya EHVM «Postroenie krivoy vtorogo poryadka, prokhodyashchey cherez dannye tochki i kasayushchikhsya dannykh pryamykhi» [The Construction of the Curve of the Second Order Passing Through the Data Points and Data Concerning Direct]. Svid. o gosudarstvennoy registratsii programmy dlya EVM, no. 2011611961, 04.03.2011 [Certificate of state registration No. 2011611961 of 03/04/2011]. (in Russian)

26. Sal'kov N.A. Prilozhenie svojstv ciklidy Dyupena k izobreteniyam [Application of Dupin Cyclide Properties to Inventions]. Geometriya i grafika [Geometry and graphics]. 2017, V. 5, I. 4, pp. 37–43. DOI: 10.12737/ article_5a17fd233418b2.84489740. (in Russian)

27. Sal'kov N.A. Formirovanie ciklicheskih poverhnostej v kineticheskoj geometrii [The formation of cyclic surfaces in kinetic geometry]. Geometriya i grafika [Geometry and graphics]. 2017, V. 5, I. 4, pp. 24–36. DOI: 10.12737/ article_5a17fbe3680f52.30844454. (in Russian)

28. Suvorov F.M. Ob izobrazhenii voobrazhaemyh tochek i voobrazhaemyh prjamyh na ploskosti i o postroenii krivyh linij vtoroj stepeni, opredeljaemyh s pomoshh'ju vo-obrazhaemyh tochek i kasatel'nyh [About the Image of the Imagined Points and the Imagined Straight Lines on the Plane and about Plotting of the Lines of the Second Degree defined by the Imagined Points and Tangents], Kazan': Tipografija imperatorskogo Universiteta Publ., 1884. – 130 p. (in Russian)

29. 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 (in Russian). DOI: 10.12737/ article_5dce6cf7ae1d70.85408915. (in Russian)

30. V. Korotkiy. Construction of a Nine-Point Quadric Surface // Journal for Geometry and Graphics, Austria, Volume 22 (2018), No. 2, pp. 183-193.