Kassel', Germany
Two plane algebraic curves intersect at the actual intersection points of these curves’ graphs. In addition to real intersection points, algebraic curves can also have imaginary intersection points. The total number of curves intersection points is equal to the product of their orders mn. The number of imaginary intersection points can be equal to or part of mn. The position of the actual intersection points is determined by the graphs of the curves, but the imaginary intersection points do not lie on the graphs of these curves, and their position on the plane remains unclear. This work aims to determine the geometry of imaginary intersection points, introduces into consideration the concept of imaginary complement for these algebraic curves in the intersection operation, determines the form of imaginary complements, which intersect at imaginary points. The visualization of imaginary complements clarifies the curves intersection picture, and the position of the imaginary intersection points becomes expected.
real curve; conic; cubic curve; quadric; imaginary complement; complex plane; complex cubic curve, imaginary component
1. Aleksandrov P.S. Lektsii po analiticheskoy geometrii [Lectures on analytic geometry]. Moscow: Nauka Publ., 1968. 382 p. (in Russian)
2. Vygodskiy M.Ya. Spravochnik po vysshey matematike [Handbook of Higher Mathematics]. Moscow: ACT: Astrel' Publ., 2006. 991 p (in Russian).
3. Girsh A.G. Kompleksnaya geometriya - evklidova i psevdoevklidova [Complex geometry - Euclidean and pseudo-Euclidean]. Moscow: «IPTs "Maska"» Publ., 2013. 216 p. (in Russian)
4. Girsh A.G. Mnimosti v geometrii [Imaginations in geometry]. Geometriya i grafika [Geometry and Graphic]. 2014, V. 2, I. 2, pp. 3-8. DOI:https://doi.org/10.12737/5583. (in Russian)
5. Girsh A.G. Naglyadnaya mnimaya geometriya [Visual imaginary geometry]. Moscow: IPTs «Maska» Publ., 2008. 216 p. (in Russian)
6. Girsh A.G. Nachala kompleksnoy geometrii. Izbrannye zadachi po konstruktivnoy geometrii s resheniyami. Chast' II - 3D [The beginnings of complex geometry. Selected problems in constructive geometry with solutions. Part II - 3D]. Kassel' Publ., 2014. 113 p. Elektronnyy resurs. Na pravakh rukopisi: ISBN 978-5-6041815-1-5. (in Russian)
7. Girsh A.G. Nachala kompleksnoy geometrii. Sbornik zadach po kompleksnoy geometrii s resheniyami. Chast' I - 2D. [The beginnings of complex geometry. Collection of problems in complex geometry with solutions. Part I - 2D]. Kassel' Publ., 2012. 191 p. Elektronnyy resurs. Na pravakh rukopisi: ISBN 978-5-6041815-0-8. (in Russian)
8. Girsh A.G. Obobshchenie «secheniy Villarso» na poverkhnosti vrashcheniya s obrazuyushchey konikoy [Generalization of "Villarso sections" on a surface of revolution with a generating conic]. Elektronnyy zhurnal po prikladnoy geometrii [Electronic journal on applied geometry]. 2003. http://www.mai.ru/~apg/ Volume 5_n11. htm. (in Russian)
9. Girsh A.G. Okruzhnosti na kompleksnoy ploskosti [Circles on a complex plane]. Geometriya i grafika [Geometry and Graphic]. 2020, V. 8, I. 4, pp. 3-12. DOI:https://doi.org/10.12737/2308-4898-2021-8-4-3-12. (in Russian)
10. Girsh A.G. O pol'ze mnimostey v geometrii [On the Use of Imaginations in Geometry]. Geometriya i grafika [Geometry and Graphic]. 2020, V. 8, I. 2, pp. 33-40. DOI:https://doi.org/10.12737/2308-4898-2020-33-40. (in Russian)
11. Girsh A. G. Fokusy algebraicheskikh krivykh [Focuses of algebraic curves]. Geometriya i grafika [Geometry and Graphic]. 2015, V. 3, I. 3, pp. 4-17. DOI:https://doi.org/10.12737/14415. (in Russian)
12. A.G. Girsh, V.A. Geometriya i grafika [Geometry and Graphic]. 2019, V. 7, I. 3, pp. 28-35. DOI: 10.12737/ article_5dce651d80b827.49830821. (in Russian)
13. Ivanov G.S., Dmitrieva I.M. O zadachakh nachertatel'noy geometrii s mnimymi resheniyami [On the problems of descriptive geometry with imaginary solutions]. Geometriya i grafika [Geometry and Graphic]. 2015, V. 3, I. 2, pp. 3-8. DOI:https://doi.org/10.12737/12163. (in Russian)
14. Pontryagin L.S. Kubicheskaya parabola [Cubic parabola]. Kvant Publ. 1984. I. 3. (in Russian)
15. Smogorzhevskiy A.S., Stolova E.S. Spravochnik po teorii ploskikh krivykh tret'ego poryadka [Handbook on the theory of plane curves of the third order]. Moscow: FM Publ., 1961. 261 p. (in Russian)
16. Hirssn A.: Ekhtepsion of the 'Villarceau-Sektion' to Surfaces of Revolution with a Generating Sonis. Jurnal for Seometru and Graphics, 6(2000/2). r. 121-132.
