employee from 01.10.2008 until now
Russian Federation
In the articles of the Geometry and Graphics magazine devoted to the properties of the Dupin cyclide, the construction of a conic – ellipse, hyperbola and parabola – using the properties of the cyclide was considered. At the same time, the center of the transformation was located on a straight line connecting the centers of the two base circles, and its location on such a straight line was negotiated separately and was located as the center of homology. To construct a parabola, it was necessary to take a straight line instead of the second circle, and the center of the transformation – the center of homology – had to be located at the intersection point of a straight line passing through the center of the first circle perpendicular to the second circle-a straight line with the first circle. Two different parabolas were obtained as a result of the transformation. In this paper, it is proved that if we take the center of correspondence that does not belong to a circle, we get other second–order curves - ellipses and hyperbolas. The construction of an ellipse is geometrically proved. To do this, the center of correspondence must lie on a straight line connecting the centers of the circles, but outside the actual circle. Several examples are considered. If the center of correspondence is inside the circle, we will have a hyperbola. Thus, having initially given only one con-figuration from a straight line and a circle, it is possible to obtain all conics: ellipses, parabolas, and hyperbolas, passing into one another. The proposed scheme for constructing conics can be used for computer drawing of all conics, which is more convenient than with the available options sewn into today's graphical drawing systems.
geometry, descriptive geometry, higher education, geometric education
1. Babakov V.V. Proektirovanie poverxnostej krivy`mi vtorogo poryadka v samoletostroenii [Designing surfaces with second-order curves in aircraft construction]. Moscow, Mashinostroenie Publ., 1969. 124 p. (in Russian)
2. Beglov I.A. Matematicheskoe opisanie metoda vrascheniya tochki vokrug krivolineynoy osi vtorogo poryadka [Tekst] / I.A. Beglov, V.V. Rustamyan, I.V. Antonova // Geometriya i grafika. - 2018. - T. 6. - № 4. - S. 39- 46. - DOI:https://doi.org/10.12737/article_5c21f6e832b4d2.25216268.
3. Beloguzhev V.A. Proektivnye sposoby postroeniya osnovnyh parametrov krivyh vtorogo poryadka, zadannyh polnym polyarnym sootvetstviem i odnim iz ih elementov [Tekst] / V.A. Beloguzhev // Voprosy nachertatel'noy geometrii i inzhenernoy grafiki: nauchnye trudy. - Tashkent, Izd-vo «FAN» Uzbekskoy SSR, 1966. - Vyp. 39. - S. 10-21.
4. Berger E.G. K sintezu mehanizmov dlya ogibaniya konicheskih secheniy metodom proektivnoy geometrii / E.G. Berger // Prikladnaya geometriya i inzhenernaya grafika: mezhvedomstvennyy respublikanskiy nauchn. sb. - Kiev, izd-vo «Budivel'nik», 1973. - Vyp. 16. - S. 110-113.
5. Voloshinov D.V. Edinyy konstruktivnyy algoritm postroeniya fokusov krivyh vtorogo poryadka obrazov [Tekst] / D.V. Voloshinov // Geometriya i grafika. - 2018. - T. 6. - № 2. - S. 47-54. - DOI: 10.12737/ article_5b559dc3551f95.26045830.
6. Girsh A.G. Vzaimnye zadachi s konikami [Tekst] / A.G. Girsh // Geometriya i grafika. - 2022. - T. 8. - № 1. - C. 4-17. DOI:https://doi.org/10.12737/2308-4898-2020-15-24.
7. Girsh A.G. Fokusy algebraicheskih krivyh [Tekst] / A.G. Girsh // Geometriya i grafika. - 2015. - T. 3. - № 3. - C. 4-17. DOI:https://doi.org/10.12737/14415.
8. Grafskiy O.A. Ob ustanovlenii vzaimnoy svyazi ryada i puchka vtorogo poryadka [Tekst] / O.A. Grafskiy // Geometriya i grafika. - 2016. - T. 4. - № 2. - C. 8-18. - DOI: 10. 12737/19828.
9. Grafskiy O.A. Osobennosti svoystv paraboly pri ee modelirovanii [Tekst] / O.A. Grafskiy, Yu.V. Ponomarchuk, V.V. Suric // Geometriya i grafika. - 2018. - T. 6. - № 2. - S. 63-77. - DOI: 10.12737/ article_5b55a16b547678.01517798.
10. Ermakova V.A. O kasanii krivyh 2-go poryadka / V.A. Ermakova // Kibernetika grafiki i prikladnaya geometriya poverhnostey: sb. statey pod red. I.I. Kotova. Trudy instituta, vyp. 2. - M.: Izd-vo MAI, 1968. - S. 77-81.
11. Korotkiy V.A. Gomologiya dvuh konicheskih secheniy / V.A. Korotkiy // Sovershenstvovanie podgotovki uchaschihsya i studentov v oblasti grafiki, konstruirovaniya i standartizacii: mezhvuz. nauch.-metod. sb. - Saratov: SGTU, 2012. - S. 27-33.
12. Korotkiy V.A. Graficheskie algoritmy rekonstrukcii krivoy vtorogo poryadka, zadannoy mnimymi elementami [Tekst] / V.A. Korotkiy, A.G. Girsh // Geometriya i grafika. - 2016. - T. 4. - № 4. - C. 19-30. - DOI:https://doi.org/10.12737/22840. (in Russian)
13. Korotkiy V.A. Krivye vtorogo poryadka v zadachah formoobrazovaniya arhitekturnyh obolochek / V.A. Korotkiy, E.A. Usmanova // Izvestiya VUZov. Seriya «Stroitel'stvo». - 2014. - № 9-10 (669-670). - S. 101-107.
14. Korotkiy V.A. Krivye vtorogo poryadka na ekrane komp'yutera [Tekst] / V.A. Korotkiy, E.A. Usmanova // Geometriya i grafika. - 2018. - T. 6. - № 2. - S. 101-113. - DOI: 10.12737/ article_5b55a829cee6c0.74112002.
15. Korotkiy V.A. Primenenie krivyh vtorogo poryadka dlya konstruirovaniya gladkih karkasno-setchatyh poverhnostey / V.A. Korotkiy, E.A. Usmanova // Vestnik YuUrGU. Seriya «Stroitel'stvo i arhitektura». - 2014. - T. 14. № 3. - S. 45-48.
16. Korotkiy V.A. Proektivnoe sootvetstvie puchkov konicheskih secheniy / V.A. Korotkiy // Informacionnye tehnologii i tehnicheskiy dizayn v professional'nom obrazovanii i promyshlennosti: sb. mater. 5-y Vserosc. nauch.-prakt. konf. s mezhdunarodnym uchastiem. - Novosibirsk: NGTU, 2013. - S. 49-56.
17. Korotkiy V.A. Soprikosnovenie konik / V.A. Korotkiy // Sovershenstvovanie podgotovki uchaschihsya i studentov v oblasti grafiki, konstruirovaniya i standartizacii: mezhvuz. nauch.-metod. sb. - Saratov: SGTU, 2011. - S. 78-82.
18. Korotkiy V.A. Formoobrazovanie liniy i poverhnostey na osnove krivyh vtorogo poryadka v komp'yuternom geometricheskom modelirovanii: 05.01.01 «Inzhenernaya geometriya i komp'yuternaya grafika»: avtoreferat dissertacii na soiskanie uchenoy stepeni doktora tehnicheskih nauk [Tekst] / V.A. Korotkiy; Nizhegorodskiy gosudarstvennyy arhitekturno-stroitel'nyy universitet. - Nizhniy Novgorod, 2018. - 38 s.
19. Korotkiy V.A. Ellipticheskiy kupol na treugol'nom ili chetyrehugol'nom fundamente / V.A. Korotkiy // Privolzhskiy nauchnyy zhurnal. - 2015. - № 1. - S. 96-102.
20. Sal'kov N.A. Grafo-analiticheskoe reshenie nekotoryh chastnyh zadach kvadratichnogo programmirovaniya [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2014. - T. 2. - № 1. - S. 3-8. - DOI:https://doi.org/10.12737/3842.
21. Sal'kov N.A. Prilozhenie svoystv ciklidy Dyupena k izobreteniyam [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2017. - T. 5. - № 4. - C. 37-43. - DOI:https://doi.org/10.12737/article_5a17fd233418b2.84489740.
22. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch. 1 [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2015. - T. 3. - № 1. - S. 16-25. - DOI:https://doi.org/10.12737/10454.
23. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch. 2 [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2015. - T. 3. - № 2. - S. 9-23. - DOI:https://doi.org/10.12737/12164.
24. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch. 3: sopryazheniya [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2015. - T. 3. - № 4. - S. 3-14. - DOI:https://doi.org/10.12737/17345.
25. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch. 4: prilozheniya [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2016. - T. 4. - № 1. - S. 21-32. - DOI:https://doi.org/10.12737/17347.
26. Sal'kov N.A. Ciklida Dyupena i krivye vtorogo poryadka. Ch. 1. [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2016. - T. 4. - № 2. - S. 19-28. - DOI:https://doi.org/10.12737/19829.
27. Sal'kov N.A. Ciklida Dyupena i krivye vtorogo poryadka. Ch. 2 [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2016. - T. 4. - № 3. - S. 17-28. - DOI:https://doi.org/10.12737/21530.
28. Sal'kov N.A. Ellips: kasatel'naya i normal' [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2013. - T. 1. - № 1. - C. 35-37. - DOI:https://doi.org/10.12737/470.
29. Svidetel'stvo o gosudarstvennoy registracii programmy dlya EVM № 2020616015 Rossiyskaya Federaciya. Giperbola: № 2020612357: zayavl. 04.03.2020: opubl. 05.06.2020 [Tekst] / N.A. Sal'kov, D. V. Voloshinov; zayavitel' Federal'noe gosudarstvennoe byudzhetnoe obrazovatel'noe uchrezhdenie vysshego obrazovaniya «Sankt-Peterburgskiy gosudarstvennyy universitet telekommunikaciy im. prof. M.A. Bonch-Bruevicha».
30. Svidetel'stvo o gosudarstvennoy registracii programmy dlya EVM № 2020614640 Rossiyskaya Federaciya. Parabola: № 2020612401: zayavl. 04.03.2020: opubl. 20.04.2020 [Tekst] / N.A. Sal'kov, D. V. Voloshinov; zayavitel' Federal'noe gosudarstvennoe byudzhetnoe obrazovatel'noe uchrezhdenie vysshego obrazovaniya «Sankt-Peterburgskiy gosudarstvennyy universitet telekommunikaciy im. prof. M.A. Bonch-Bruevicha».
31. Svidetel'stvo o gosudarstvennoy registracii programmy dlya EVM № 2020616140 Rossiyskaya Federaciya. Ellips: № 2020612388: zayavl. 04.03.2020: opubl. 10.06.2020 [Tekst] / N.A. Sal'kov, D. V. Voloshinov; zayavitel' Federal'noe gosudarstvennoe byudzhetnoe obrazovatel'noe uchrezhdenie vysshego obrazovaniya «Sankt-Peterburgskiy gosudarstvennyy universitet telekommunikaciy im. prof. M.A. Bonch-Bruevicha».
32. Heyfec A.L. Koniki kak secheniya kvadrik ploskost'yu (obobschennaya teorema Dandelena) [Tekst] / A.L. Heyfec // Geometriya i grafika. - 2017. - T. 5. - № 2. - C. 45-58. DOI:https://doi.org/10.12737/article_5953f32172a8d8.94863595.
33. Salkov N.A. Setting of the Dupin cyclide by three straight lines and sphere / N.A. Salkov. - Text: direct // IOP Conf. Series: Journal of Physics: Conf. Ser. 1791 (2021) 012060. doihttps://doi.org/10.1088/1742-6596/1791/1/012060.
