Habarovsk, Khabarovsk, Russian Federation
Habarovsk, Khabarovsk, Russian Federation
Habarovsk, Khabarovsk, Russian Federation
Electrostatic fields have been most fully studied as special cases of electromagnetic field. They are created by a set of charged bodies that are considered immovable in relation to the observer and unchanged in time [1–3; 19; 20; 27]. Since any field is characterized by basic quantities, then such quantities for electrostatic fields are strength E and potential ϕ. Therefore, geometrically, such fields are characterized by a combination of force and equipotential lines. These fields were considered in the thesis of N.P. Anikeeva [1]. In particular, the author notes that in the case of dissimilar equal charges "... families of force and equipotential lines make up two orthogonal bundles of circles" [1, p. 59]. However, it is necessary to clarify that each "force" circle gj represents by itself not one but two lines of force, which emanate from a positive charge and terminate on a negative one. A similar review on the work [1] can be done with respect to the picture of two equal positive charges’ electrostatic field. Here the author considers a family ui of equipotential lines, which are Cassini ovals. It is truly said that these ovals belong to the fourth-order bicircular curves of genre 1 (have two double imaginary points, which are cyclic). But these ovals’ family includes one curve of zero genre — the Bernoulli lemniscate; it has three double points (two of them are the same cyclic ones, and one is real, which coincides with the origin of coordinates). In addition, it has been noted that "... the lines of current are equilateral hyperbolas gj» [1, p. 63]. However, clarification is also required here. The lines of force exit from each point charge and each line has two opposite directions. One such line of "double direction" forms only one branch of an equilateral hyperbola. A similar set of branches of equilateral hyperbolas also emanates from the second charge.
electrostatic fields, power and equipotential lines, cyclic points, involution, harmonism, imaginary elements.
Целью данной статьи является выработка рекомендаций необходимых для построения и анализа электростатических полей, которые особенно необходимы при их теоретическом изучении.
1. Anikeeva N.P. Geometricheskoe modelirovanie kartiny ehlektricheskogo polya v kamere osazhdeniya iznosostojkih pokrytij. Kand. Diss. [Geometrical modeling of a picture of electric field in the camera of sedimentation of wearproof coverings. Cand. Diss.]. Moscow, 1998. 127 p. (in Russian)
2. Bessonov L.A. Teoreticheskie osnovy ehlektrotekhniki. Ehlektromagnitnoe pole [Theoretical bases of electrical equipment. Electromagnetic field]. Moscow, Gardaraki Publ., 2001. 317 p. (in Russian)
3. Gershenzon E.M., Malov N.N. Ehlektrodinamika [Ehlektrodinamika]. Moscow, Akademiya Publ., 2002. 352 p. (in Russian)
4. Hirsch A. Kriteriy mnimosti konik pri razlichnom ih zadanii [Criterion of ostensibility the skate at their various task]. Mezhvuzovskiy tematicheskiy sbornik nauchnych trudov "Nachertatelnaya geometriya i mashinnaya grafika v prakticheskom reshenii inzhenernyh zadach" [Descriptive geometry and machine graphics in the practical solution of engineering tasks: Interuniversity thematic collection of scientific works]. Omsk, 1986. pp. 51-54. (in Russian)
5. Girsh A.G. Mnimosti v geometrii [Imaginaries in Geometry]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 2, pp. 3-8. (in Russian)
6. Hirsch A.G. Tochki peresecheniya i obshchie kasatelnye dvuh okruzhnostey [Points of intersection and general tangents of two circles]. Nachertatelnaya geometriya i mashinnaya grafika v prakticheskom reshenii inzhenernyh zadach: Mezhvuzovskiy tematicheskiy [Descriptive geometry and machine graphics in the practical solution of engineering tasks: Interuniversity thematic collection of scientific works]. Omsk, 1987. pp. 53-57. (in Russian)
7. Girsh A.G. Fokusy algebraicheskikh krivykh [Focuses algebraic curves]. Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 3, pp. 4-17. (in Russian)
8. Glagolev N.A. Proektivnaja geometrija [Projective geometry]. Moscow, Vysshaya shkola Publ., 1963. 344 p. (in Russian)
9. Grafskij O.A., Chernyavskaya S.N. Konstruktivnye i analiticheskie issledovaniya mnimyh ehlementov [Constructive and analytical researches of imaginary elements]. Problemy i perspektivy razvitiya nauki i obrazovaniya v XXI veke: Materialy Mezhdunarodnoj (zaochnoj) nauchno-prakticheskoj konf. [Problems and prospects for the development of science and education in the 21st century: Materials of the International (correspondence) scientific and practical conference]. Sofiya, K"shcha "SORoS Publ., 2017, pp. 133-141. (in Russian)
10. Grafskij O.A., Usmanov A.V., Holodilov A.A. Grafoanaliticheskie issledovaniya involyucii [Graphic-analytical researches of involution]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 1, pp. 3-11. (in Russian)
11. Grafskiy O.A. Modelirovanie mnimykh elementov na ploskosti [Simulation imaginary elements in the plane]. Khabarovsk, DVGUPS Publ., 2004. 162 p. (in Russian)
12. Grafskij O.A. Ob ustanovlenii vzaimnoj svyazi ryada i puchka vtorogo poryadka [About establishment of an inter-connection of a row and bunch of the second order]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 2, pp. 8-18. (in Russian)
13. Grafskij O.A., Ponomarchuk Yu. V. Okruzhnost koordinatnoy ploskosti [Circle of the coordinate plane Natural and technical science]. Estestvennie i tehnicheskie nauki [Natural and technical science]. 2017, I. 1 (103), pp. 131-136. (in Russian)
14. Grafskiy O.A. Teoretiko-konstruktivnye problemy modelirovaniya mnimykh elementov v nachertatel'noy geometrii i ee prilozheniyakh. Dokt. Diss. [Teoretiko-construktive problems of modeling of imaginary elements in descriptive geometry and its applications. Doct. Diss.]. Moscow, 2004. 406 p. (in Russian)
15. Ivanov G.S. Konstruirovanie tekhnicheskih poverhnostej (matematicheskoe modelirovanie na osnove nelinejnyh preobrazovanij) [Designing of technical surfaces (mathematical modeling on the basis of nonlinear transformations)]. Moscow, Mashinostroenie Publ, 1987. 192 p. (in Russian)
16. Ivanov G.S, Dmitrieva, I.M. O zadachah nachertatelnoy geometrii s mnimymi resheniyami [About the Tasks of Descriptive Geometry With Imaginary Solutions]. Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 2, pp. 3-8. (in Russian)
17. Ivanov G. Perspektivy nachertatelnoy geometrii kak uchebnoy distsipliny [Descriptive geometry prospects as educational subject]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 1, pp. 26-27. (in Russian)
18. Ivanov G.S., Dmitrieva I.M., Princip dvojstvennosti - teoreticheskaya baza vzaimosvyazi sinteticheskih i analiticheskih sposobov resheniya geometricheskih zadach [The principle of a duality - theoretical base of interrelation of synthetic and analytical ways of the solution of geometrical tasks]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 3, pp. 3-10. (in Russian)
19. Kostryukov A.V. Geometricheskoe modelirovanie processa formirovaniya poverhnostej pri osazhdenii tonkoplenochnyh pokrytij. Kand. Diss. [Geometrical modeling of process of formation of surfaces at sedimentation of thin-film coverings. Cand. Diss.]. Moscow, 1992. 191 p. (in Russian)
20. Kusebaev U.K. Konstruirovanie special'nyh geometricheskih modelej dlya opisaniya ehlektricheskogo polya LEHP. Kand. Diss. [Designing of special geometrical models for the description of electric field of the high voltage line. Cand. Diss.]. Kiev, 1990. 125 p. (in Russian)
21. Miln-Tomson L.M. Teoreticheskaya gidrodinamika [Theoretical hydrodynamics]. Moscow, Mir Publ., 1964. 655 p. (in Russian)
22. Ponomarchuk YU.V., Saharova N.A. Issledovanie okruzhnosti v dekartovoj ploskosti [Research of a circle in the Cartesian plane]. Nauka, obrazovanie, innovacii: aprobaciya rezul'tatov issledovanij: Materialy (zaochnoj) nauchno-prakticheskoj konferencii [Science, education, innovations: approbation of research results: Materials (correspondence) scientific-practical conference]. Praga, 2017, pp. 20-27. (in Russian)
23. Savelov A.A. Ploskie krivye. Sistematika, svoystva, primeneniya: Spravochnoe rukovodstvo [Flat curves. Systematization, properties, applications: Reference guide]. Moscow, RHD Publ., 2002. 294 p. (in Russian)
24. Savel´ev Y.A. Grafika mnimykh chisel [Graphics of imaginary numbers]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 1, pp. 22-23. (in Russian)
25. Sal´kov N.A. Mesto nachertatel'noj geometrii v sisteme geometricheskogo obrazovaniya tekhnicheskih vuzov [Place of descriptive geometry in system of geometrical formation of technical colleges]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 3, pp. 53-61. (in Russian)
26. Sal´kov N.A. Nachertatel'naya geometriya - baza dlja geometrii analiticheskoj [Geometry As the Basis for Analytical Geometry]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 1, pp. 44-54. (in Russian)
27. Tatur T.A. Osnovy teorii ehlektromagnitnogo polya [Bases of the theory of an electromagnetic field ]. Moscow, Vyssh. shk. Publ., 1989. 271 p. (in Russian)
28. Poncelet J.-V. Traitè des propriètès projectives des figures // Applications ďAnalyse et des Gèomètrie. Paris, 1862. V. 1. 563 p., 1864. V. 2. 602 p.
