Moskva, Moscow, Russian Federation
from 01.10.2008 until now
Russian Federation
Moskva, Moscow, Russian Federation
Loci of points (LOP) equally spaced from two given geometrical figures are considered. Has been proposed a method, giving the possibility to systematize the loci, and the key to their study. The following options have been considered. A locus equidistant from N point and l straight line. N belongs to l. We have a plane that is perpendicular to l and passing through N. N does not belong to l – parabolic cylinder. A locus equidistant from F point and a plane. In the general case, we have a paraboloid of revolution. The F point belongs to the given plane. We get a straight line perpendicular to the plane and passing through the F point. A locus equidistant from a point and a sphere. The point coincides with the sphere center. We get the sphere with a radius of 0.5 R. The point lies on the sphere. We get the straight line passing through the sphere center and the point. The point does not coincide with the sphere center, but is inside the sphere. We get the ellipsoid. The point is outside the sphere. We have parted hyperboloid of rotation. A locus equidistant from a point and a cylindrical surface. The point lies on the cylindrical surface’s axis. We get the surface of revolution which generatix is a parabola. The point lies on the generatrix of the cylindrical surface of rotation. We get a straight line, perpendicular to that generatrix and passing through the cylinder axis. The point does not lie on the axis, but is located inside the cylindrical surface. We get the surface with a horizontal sketch line – the ellipse, and a front sketch lines – two different parabolas. The point is outside the cylindrical surface. A locus consists of two surfaces. The one with the positive Gaussian curvature, and the other – with the negative one.
geometry, descriptive geometry, loci, locus, analytical geometry.
Начертательная геометрия «является наивысшим средством развития той таинственной и мало поддающейся изучению точными науками способности человеческого духа, которая зовется воображением, и которая является ступенью к другой царственной способности — фантазии, без которой почти не совершаются великие открытия и изобретения».
Рынин Н.А.*1[19, с. 153]
1. Volkov V.Ja., Jurkov V.Ju., Panchuk K.L., Kajgorodceva N.V. Sbornik zadach i uprazhnenij po nachertatel'noj geometrii (k uchebniku «Kurs nachertatel'noj geometrii na osnove geometricheskogo modelirovanija») [Collection of problems and exercises on descriptive geometry (to the textbook "the Course of descriptive geometry on the basis of geometric modeling")]. Omsk: SIBADI Publ., 2010. 74 p. (in Russian)
2. Girsh A.G. Kak reshat' zadachu. Metodicheskie ukazanija po resheniju zadach povyshennoj slozhnosti [How to solve the problem. Guidelines for solving problems of high complexity]. Omsk: SIBADI Publ., 1986. 36 p. (in Russian).
3. Eliseev N.A. Jetjudy po nachertatel'noj geometrii professora D.I. Kargina. Sovershenstvovanie podgotovki uchashhihsja i studentov v oblasti grafiki, konstruirovanija i standartizacii [Essays on descriptive geometry by Professor D.I. Kargin. Improving the training of pupils and students in the field of graphics, design and standardization]. Mezhvuzovskij nauchno-metodicheskij sbornik [Interuniversity scientific-methodical collection]. Saratov: SGTU Publ., 2004, pp. 56-58. (in Russian).
4. Ivanov G.S. Nachertatel'naja geometrija [Descriptive geometry]. Moscow, FGBOU VPO MGUL Publ., 2012. 340 p. (in Russian).
5. Ivanov G.S. Teoreticheskie osnovy nachertatel'noj geometrii [Theoretical foundations of descriptive geometry]. Moscow, Mashinostroenie Publ., 1998. 458 p. (in Russian).
6. Kargin D.I. E`tjudy po nachertatel'noj geometrii. Geometricheskije mesta [Studies on descriptive geometry. Geometric space]. PFA RAN Publ. (in Russian).
7. Obukhova V.S. Poetapnoe modelirovanie tekhnicheskikh poverkhnostey [Incremental modeling of technical surfaces]. Referativnaja informatsija o zakonchenny`kh nauchno-issledovatel'skikh rabotakh v vuzakh Ukrainskoy SSR: Prikladnaya geometrija i inzhenernaja grafika [Abstracts information on completed research projects in the Ukrainian SSR universities: Applied geometry and engineering graphics]. Kiev, Vishcha shkola Publ., 1977, pp. 5-6. (in Russian).
8. Pavlov V.E., Tarasov B.F. Dmitrij Ivanovich Kargin. 1880-1949 [Dmitry Ivanovich Kargin. 1880-1949]. St. Petersburg, Nauka Publ., 1998. 272 p. (in Russian).
9. Posvjanskij A.D. Pjat`desjat zadach povy`shennoj trudnosti [Fifty-task advanced]. Kalinin: KPI Publ., 1970. 41 p. (in Russian).
10. Sal'kov N.A. Geometricheskoe modelirovanie i nachertatel'naja geometrija [Geometric modeling and descriptive geometry]. Geometrija i grafika [Geometry and Graphics]. 2016, V. 4, I. 4, pp. 31-40. DOI:https://doi.org/10.12737/22841. (in Russian).
11. Sal'kov N.A. Mesto nachertatel'noj geometrii v sisteme geometricheskogo obrazovanija tehnicheskih vuzov [Place of descriptive geometry in the geometric system of education of technical universities]. Geometrija i grafika [Geometry and Graphics]. 2016, V. 4, I. 3, pp. 53-61. DOI:https://doi.org/10.12737/21534. (in Russian).
12. Sal'kov N.A. Metody parametricheskoj geometrii v modelirovanii avtomobil'nyh dorog [Methods of parametric geometry modelling of roads]. Zhurnal estestvennonauchnyh issledovanij [Journal of Natural Science Studies]. 2016, V. 1, I. 4, pp. 1-1. DOI:https://doi.org/10.12737/22143. (in Russian).
13. Sal'kov N.A. Nachertatel'naya geometriya - baza dlya komp'yuternoy grafiki [Descriptive Geometry - the basis for computer graphics]. Geometrija i grafika [Geometry and Graphics]. 2016, V. 4, I. 2, pp. 37-47. DOI:https://doi.org/10.12737/19832. (in Russian).
14. Salkov N.A. Nachertatel`naja geometrija. Bazovy`j kurs [Descriptive geometry: Base kurs]. Moscow, INFRA-M Publ., 2013. 184 p. (in Russian).
15. Sal'kov N.A. Nachertatel`naja geometrija - teorija izobrazhenij [Descriptive geometry - theory of images]. Geometrija i grafika [Geometry and Graphics]. 2016, V. 4, I. 4, pp. 41-47. DOI:https://doi.org/10.12737/22842.
16. Sal'kov N.A., Vyshnepol'skij V.I. O vozrastajushhej roli geometrii [On the growing role of geometry]. Zhurnal estestvennonauchny`h issledovanij [Journal of Natural Science Studies]. 2017, V. 2, I. 2, pp. 53-61. Available at: https://naukaru.editorum.ru/ru/nauka/article/16413/view
17. Seregin V.I., Ivanov G.S., Dmitrieva I.M., Murav'ev K.A. Mezhdisciplinarny`e svjazi nachertatel'noj geometrii i smezhny`h razdelov vy`sshej matematiki [Interdisciplinary team of descriptive geometry and related topics of higher mathematics]. Geometrija i grafika [Geometry and Graphics]. 2013, V. 1, I. 4, pp. 8-12.
18. Tarasov B.F. Valerian Ivanovich Kurdjumov [Valerian Ivanovich Kurdiumov]. St. Petersburg, Nauka Publ., 1997. 231 p.
19. Tarasov B.F. Nikolaj Alekseevich Rynin. 1877-1942 [Nikolai Rynin. 1877-1942]. Nauka Publ., 1990. 168 p.
20. Tarasov B.F. Jakov Aleksandrovich Sevast`janov [Yakov Aleksandrovich Sevastyanov]. St. Petersburg, Nauka Publ., 1995. 215 p.
21. Teoreticheskie osnovy` formirovanija modelej poverkhnostej [Theoretical bases of formation of surface models]. Moscow, MAI Publ., 1985. (in Russian).
22. Umbetov N.S. Konstruirovanie ekvipotentsial'noy poverkhnosti [Construction equipotential surface]. Geometrija i grafika [Geometry and Graphics]. 2013, V. 1, I. 1, pp. 11-14. DOI:https://doi.org/10.12737/2075. (in Russian).
23. Shal' M. Istoricheskiy obzor proiskhozhdeniya i razvitiya geometricheskikh metodov [Historical overview of the origins and development of geometric methods]. Moscow, 1883. (in Russian).
