Kassel', Germany
The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.
circle: real, imaginary, degenerated; real circle + imaginary complement; imaginary circle + actual complement; zero circle + left and right isotropies; right angle: real, imaginary; radical axis; radical center; sphere
1. Hadamard J. Elementarnaya geometriya. Chast’ I [Elementary geometry]. Moscow-Leningrad: Uchpedgiz Publ., 1948. 608 p. (in Russian)
2. Voloshinov D.V. Algorimetricheskij kompleks dlya reshenia zadach s kvadrikami s primeneniem mnimyh geometricheskih obrazov [Algorithmic Complex for Solving of Problemswith Quadrics Using Imaginary Geometric Images]. Geometriya i grafika [Geometry and Graphics]. 2020, V. 8, I. 2, pp. 3-32. DOIhttps://doi.org/10.12737/2308-4898-2020-3-32. (in Russian)
3. Voloshinov D.V. Vizual'no-graficheskoe proektirovanie edinoj konstruktivnoj modeli dlya resheniya analogov zadachi Apolloniya s uchetom mnimyh geometricheskih obrazov [Visual and graphic design of a single constructive model for solving analogues of the Apollonius problem taking into account imaginary geometric images]. Geometriya i grafika [Geometry and Graphics]. 2018, V. 6, I. 2, pp. 23-46. DOIhttps://doi.org/10.12737/article_5b559c70becf44.21848537. (in Russian)
4. Voloshinov D.V. Edinyj konstruktivnyj algoritm postroeniya fokusov krivyh vtorogo poryadka [A single constructive algorithm for constructing the foci of second-order curves]. Geometriya i grafika [Geometry and Graphics]. 2018, V. 6, I. 2, pp. 47-54. DOI:https://doi.org/10.12737/article_5b559dc3551f95.26045830. (in Russian)
5. Girsh A.G. Kompleksnaya geometriya - evklidova i psevdoevklidova [Complex geometry - Euclidean and pseudo-Euclidean]. Moscow: Maska Publ., 2013. 216 p. (in Russian)
6. Girsh A.G. Mnimosti v geometrii [Imaginations in geometry]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 2, pp. 3-8. - DOI:https://doi.org/10.12737/5583. (in Russian)
7. Girsh A.G. Naglyadnaya mnimaya geometriya [Visual imaginary geometry]. Moscow: Maska Publ., 2008. 216 p. (in Russian)
8. Girsh A.G. Nachala kompleksnoj geometrii. Izbrannye zadachi kompleksnoj geometrii s resheniyami. Chast' II - 3D [The beginning of complex geometry. Selected problems of complex geometry with solutions. Part II - 3D]. Kassel' Publ., 2014. 112 p. (in Russian)
9. Girsh A.G. Nachala kompleksnoj geometrii. Sbornik zadach po kompleksnoj geometrii s resheniyami. Chast' I - 2D [The beginning of complex geometry. Collection of problems in complex geometry with solutions. Part I - 2D]. Kassel' Publ., 2012. 191 p. (in Russian)
10. Girsch A.G. Novye zadachi nachertatel'noj geometrii [New Descriptive Geometry Problems]. Geometriya i grafika [Geometry and graphics]. 2019, V. 7, I. 4, pp. 18-33. DOIhttps://doi.org/10.12737/2308-4898-2020-18-33. (in Russian)
11. Hirsch A.G. O pol′ze mnimostej v geometrii [In Favor of Imaginaries in Geometry]. 2020, V. 8, I. 2, pp. 3-32. DOIhttps://doi.org/10.12737/2308-4898-2020-33-40. (in Russian)
12. Girsch A.G., Korotky V.A. Graficheskie algoritmy rekonstrukcii krivoj vtorogo poryadka, zadannoj mnimymi elementami [Graphic Algorithms for Reconstructing a Second Order Curve Given by Imaginary Elements]. Geometriya I grafika [Geometry and graphics]. 2016, V. 4, I. 4, pp. 19-30. DOI:https://doi.org/10.12737/22840. (in Russian)
13. Girsh A.G., Korotkij V.A. Mnimye tochki v dekartovoj sisteme koordinat [Imaginary points in a Cartesian coordinate system]. Geometriya i grafika [Geometry and Gra-phics]. 2019, V. 7, I. 3, pp. 28-35. DOI:https://doi.org/10.12737/artcle_5dce651d80b827.49830821. (in Russian)
14. Ivanov G.S., Dmitrieva I.M. O zadachah nachertatel'noj geometrii s mnimymi resheniyami [On descriptive geometry problems with imaginary solutions]. Geometriya i grafika [Geometry and graphics]. 2015, V. 3, I. 2, pp. 3-8. DOI:https://doi.org/10.12737/12163. (in Russian)
15. Korotkij V.A. Mnimye pryamye v dekartovoj sisteme koordinat [Imaginary straight lines in a Cartesian coordinate system]. Geometriya i grafika [Geometry and Graphics]. 2019, V. 7, I. 3, pp. 28-35. DOIhttps://doi.org/10.12737/artcle_5dce651d80b827.49830821. (in Russian)
16. Ponarin Ya. P. Elementarnaya geometriya [Elementary geometry]. Moscow: MCCNMO Publ., 2006. 256 p. (in Russian)
17. Prasolov V.V. Zadachi-po-planimetrii [Tasks on planimetry]. ICMMO Publ., 2001. 640 p. (in Russian)
18. Programma dlja JeVM «Postroenie krivoj vtorogo porjadka, prohodjashhej cherez dannye tochki i kasajushhejsja dannyh prjamyh» [Computer program "Construction of a curve of the second order passing through the data points and touching the data lines"]. Svidetel'stvo o gosudarstvennoj registracii № 2011611961 ot 04.03.2011 g. [Certificate of state registration № 2011611961 dated 03/04/2011]. (in Russian)
19. Yaglom I.M. Kompleksnyye chisla i ikh primeneniye v geometrii [Complex numbers and heir application in geometry]. Moscow, Editorial URSS Publ., 2004. 192 p. (in Russian)
20. Duden Rechnen und Mathematik: - Mannheim, Wien, Zürich: Dudenverlag, 1985.
21. Hirsch, A. Extension of the 'Villarceau-Sektion' to Surfaces of Revolution with a Generating Conic // Jurnal for Geometriy and Graphics, V. 6 (2000), I. 2, pp. 121-132.
22. Stachel H.: Remarks on A. Hirsch's Paper conserning Villatceau-Sections. J. Geometry Graphics 6, pp. 133-139 (2002).
