Moskva, Moscow, Russian Federation
Moskva, Moscow, Russian Federation
Moskva, Moscow, Russian Federation
In this paper are studied surfaces which are loci of points (LOP) equally spaced from a point and a conical surface under a variety of the point and conical surface’ mutual arrangement. Mathematical models of such surfaces are studied, and mathematical analysis of their properties is performed, as well as 3D models of considered surfaces are constructed. Possible cases of mutual arrangement for the point and the conical surface: • the point is at the conical surface’s vertex; • the point is on the conical surface; • the point is inside the conical surface: –– on the axis, –– not on the axis; • the point is outside the conical surface. The point is on the vertex of the conical surface Γ — the obtained conical surface Ω has the same vertex, whose generatrixes are perpendicular to the generatrixes of the surface Γ. The point is on the conical surface Γ — LOP equally spaced from the surface Γ and the point O separates into a straight-line l and a surface Φ of 4th order. The line l is located in the axial plane passing through the point O and is perpendicular to the generatrix of the conical surface Γ. Obtained surface Φ has a symmetry plane passing through the axis of the conical surface Γ and the point O. Many sections of the obtained surface Φ are Pascal snails. The point is inside the conical surface on the axis. Obtained surface α is a rotation surface, and the axis z is its axis of rotation. All the sections of the surface by planes perpendicular to the axis z are circles. Point is outside the conical surface. A very interesting surface Ω has been obtained, with the following properties: the surface Ω has a support plane, which is tangent to the surface Ω on a hyperbole; the surface Ω has 2 symmetry planes; there are a circle, parabola and Pascal’s snail among the surface Ω sections. In this paper have been considered analogues between surfaces of LOP equally spaced from the cylindrical surface and the point, and from the conical surface and the point.
geometry, descriptive geometry, locus, LOP, analytic geometry, conical surface, surface construction, Pascal snail.
Вступление
Первой известной нам работой по геометрическим местам была рукопись профессора Дмитрия Ивановича Каргина (1880–1949) «Этюды по начертательной геометрии» [8; 12]. Потом последовали работы Н.В. Наумовича (1962 г.) [15], А.Д. Посвянского (1970 г.) [17], В.С. Обуховой (1977 г.) [16], А.Г. Гирша (1986 г.) [7].
1. Aleksandrov I.I. Sbornik geometricheskih zadach na postroenie s resheniyami [Collection of geometric construction problems with solutions]. Moscow, URSS Publ., 2004. 176 p. (in Russian).
2. Volkov V.YA. Sbornik zadach i uprazhnenij po nachertatel'noj geometrii (k uchebniku «Kurs nachertatel'noj geometrii na osnove geometricheskogo modelirovaniya») [Collection of tasks and problems on descriptive geometry (for the textbook “Descriptive geometry course on the basis of geometrical modeling”)]. Omsk, SIBADI Publ., 2010. 74 p. (in Russian).
3. Vygodskij M.YA. Analiticheskaya geometriya [Analytical geometry]. Moscow, Fizmatgiz Publ., 1963. 523 p. (in Russian).
4. Vyshnepol'skij V.I. Vserossijskij studencheskij konkurs «Innovacionnye razrabotki» [Panrussian student competition “Innovative developments”]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 4, pp. 69-86. (in Russian).
5. Vyshnepol'skij V.I. Geometricheskie mesta tochek, ravnootstoyashchih ot dvuh zadannyh geometricheskih figur. CHast' 1 [Geometric locations of the points equally spaced from two given geometric figures]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 3, pp. 21-35. (in Russian).
6. Vyshnepol'skij V.I. Metodicheskie osnovy podgotovki i provedeniya olimpiad po graficheskim disciplinam v vysshej shkole. Kand. Diss. [Methodical bases of preparation and holding the competitions of graphical disciplines in higher education. Cand. Diss.]. Moscow, 2000. 250 p. (in Russian).
7. Girsh A.G. Kak reshat' zadachu. Metodicheskie ukazaniya po resheniyu zadach povyshennoj slozhnosti [How to solve a problem. Methodical indications on solving problems of higher complexities]. Omsk, SIBADI Publ., 1986. 36 p. (in Russian).
8. Eliseev N.A. EHtyudy po nachertatel'noj geometrii professora D.I. Kargina. Sovershenstvovanie podgotovki uchashchihsya i studentov v oblasti grafiki, konstruirovaniya i standartizacii [Prof. Karigin’s etudes on descriptive geometry. Development of preparing students in graphics, construction and standartisation]. Saratov, SGTU Publ., 2004, pp. 56-58. (in Russian).
9. Ivanov G.S. Nachertatel'naya geometriya [Descriptive geometry]. Moscow, FGBOU VPO MGUL Publ., 2012. 340 p. (in Russian).
10. Ivanov G.S. Princip dvojstvennosti - teoreticheskaya baza vzaimosvyazi sinteticheskih i analiticheskih sposobov resheniya geometricheskih zadach [Dualism principle - theoretical base of the relationship of synthetic and analytical ways of solving geometric problems]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 5, I. 3, pp. 3-10. (in Russian).
11. Ivanov G.S. Teoreticheskie osnovy nachertatel'noj geometrii [Theoritical bases of descriptive geometry]. Moscow, Mashinostroenie Publ., 1998. 458 p. (in Russian).
12. Kargin D.I. EHtyudy po nachertatel'noj geometrii. Geometricheskie mesta [Etudes in descriptive geometry. Geometric locations]. PFA RAN Publ., p. 802. (in Russian).
13. Krivoshapko S.N. EHnciklopediya analiticheskih poverhnostej [Encyclopedia of analytical surfaces]. Moscow. LIBROKOM Publ., 2010. 560 p. (in Russian).
14. Krivoshapko S.N. Analiticheskie poverhnosti v arhitekture zdanij, konstrukcij i izdelij [Analytical surfaces in building architecture, constructions and products]. Moscow, «LIBROKOM» Publ., 2012. 328 p. (in Russian).
15. Naumovich N.V. Geometricheskie mesta v prostranstve i zadachi na postroenie [Geometric locations in space and problems on construction]. Gos. uchebno-pedagogicheskoe Publ., 1962. 152 p. (in Russian).
16. Obuhova V.S. Poehtapnoe modelirovanie tekhnicheskih poverhnostej [Step by step modeling of technical surfaces]. Referativnaya informaciya o zakonchennyh nauchno-issledovatel'skih rabotah v vuzah Ukrainskoj SSR. Prikladnaya geometriya i inzhenernaya grafika [Referative information about finished researches in the Higher educational Institutions of Ukrainian SSR. Applied geometry and engineer graphics]. Kiev: Vishcha shkola Publ., 1977, pp. 5-6. (in Russian).
17. Posvyanskij A.D. Pyat'desyat zadach povyshennoj trudnosti [Fifty problems of increased complexity]. Kalinin, KPI Publ., 1970. 41 p. (in Russian).
18. Sal'kov N.A. Nachertatel'naya geometriya: bazovyj kurs [Descriptive geometry: basis course]. Moscow. INFRA-M Publ., 2013. 184 p. (in Russian).
19. Sal'kov N.A. Nachertatel'naya geometriya - teoriya izobrazhenij [Descriptive geometry - picture theory]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 4, pp. 41-47. (in Russian).
20. Seregin V.I. Mezhdisciplinarnye svyazi nachertatel'noj geometrii i smezhnyh razdelov vysshej matematiki [Interdiscipline connections of descriptive geometry and related sections of higher mathematics]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 3-4, pp. 8-12. (in Russian).
21. Seregin V.I. Nauchno-metodicheskie voprosy podgotovki studentov k olimpiadam po nachertatel'noj geometrii [Scientific - methodical questions on preparation of students for descriptive geometry Olympiads]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 1, pp. 73-81. (in Russian).
