student
Moskva, Moscow, Russian Federation
This article is a continuation of the study of the process of reflection of various objects from curved mirrors. So, earlier in the works [18; 20], a geometric method of constructing the results of reflections was described, which was implemented mathematically in the article [38] using the principles of analytical geometry [6; 11–14; 30]. The obtained analytical equations of the reflection results were visualized in the Wolfram Mathematica [24] program with the ability to dynamically change the parameters of the mirror and the reflected object. However, in the listed works, only cases of reflection on the plane were considered. In this study, attention is paid to a more complex case — reflection in three-dimensional space. The article considered the reflection of a point from surfaces of the second order: a cylinder, a cone, a single-cavity and double-cavity hyperboloids, a sphere, elliptical and hyperbolic paraboloids, and from a torus — a surface of the fourth order. As before, the reflection result obtained in each of the cases is accompanied by a program code for Wolfram Mathematica, which allows the reader to independently simulate the reflection process with different initial parameters. In addition, the relationships between the results obtained were analyzed — both the relationships between the results of various three-dimensional reflections, and the relationship of the results of three-dimensional reflections with the results of similar plane reflections. In particular, on the basis of this, a hypothesis was formulated about the relationship between the curvature of the Gaussian mirror and the dimension of the object obtained as a result of reflection. Based on the results of the work, conclusions were drawn and prospects for further research were outlined. One of them is to obtain an analytical mechanism for describing complex geometric surfaces using a set of simpler objects. This feature will increase the efficiency of specialists when working with reflections from complex surfaces in areas such as aircraft construction (for creating aerodynamic surfaces and air ducts), medicine [40], shipbuilding [7; 31; 42], etc.
reflection geometry, reflection in three-dimensional space, curved mirrors, parametric equations of curves, Wolfram Mathematica, the Gaussian curvature
1. Antonova I.V. Matematicheskoe opisanie vrascheniya tochki vokrug ellipticheskoy osi v nekotoryh chastnyh sluchayah [Tekst] / I.V. Antonova, I.A. Beglov, E.V. Solomonova // Geometriya i grafika. - 2019. T. 7. - №. 3 - S. 36-50. - DOI:https://doi.org/10.12737/article_5dce66dd9fb966.59423840.
2. Antonova I.V. Matematicheskoe opisanie chastnogo sluchaya kvazivrascheniya fokusa ellipsa vokrug ellipticheskoy osi [Tekst] / I.V. Antonova, E.V. Solomonova, N.S. Kadykova // Geometriya i grafika. - 2021. T. 9. - № 1. - S. 39-45. - DOI:https://doi.org/10.12737/2308-48982021-9-1-39-45.
3. Artyuhina N.K. Osnovy komp'yuternogo modelirovaniya opticheskih sistem razlichnyh klassov: uchebno-metodicheskoe posobie dlya studentov special'nosti 1-38 01 02 / N.K. Artyuhina. - Minsk: Izd-vo BNTU, 2016. 182 s.
4. 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.
5. Beglov I.A. Metod vrascheniya geometricheskih ob'ektov vokrug krivolineynoy osi [Tekst] / I.A. Beglov, V.V. Rustamyan // Geometriya i grafika. - 2017. T. 5. - № 3. - S. 45-50. - DOI:https://doi.org/10.12737/article_59bfa4 eb0bf488.99866490.
6. Beklemishev D.V. Kurs analiticheskoy geometrii i lineynoy algebry: uchebnik [Tekst] / D.V. Beklemishev 13-e izd., ispr. - SPb.: Lan', 2015. - 448 s.
7. Belov O.A. Analiz rezul'tatov ul'trazvukovogo kontrolya korpusa rybopromyslovogo sudna [Tekst] / O.A. Belov // Tehnicheskaya ekspluataciya vodnogo transporta: problemy i puti razvitiya. - 2022. № 5. - S. 5-9.
8. Belous Yu.V. Issledovanie vliyaniya cilindricheskih granic na pole parametricheskoy antenny i razrabotka sposoba vosstanovleniya formy otrazhayuschey poverhnosti [Tekst]: avtooref. dis. … kand. tehn. nauk: 01.04.06 / Yu.V. Belous. - Taganrog, 2002. - 152 s.
9. Berdyshev V.I. Approksimaciya funkciy, szhatie chislennoy informacii, prilozheniya [Tekst] / V.I. Berdyshev, L.V. Petrak. - Ekaterinburg: Izd-vo UrO RAN, 1999. - 295 s.
10. Blinova I.V. Krivye, zadannye parametricheski i v polyarnyh koordinatah [Tekst] / I.V. Blinova, I.Yu. Popov - SPb.: Izd-vo Universiteta ITMO, 2017. - 55 s.
11. Bugrov Ya.S. Vysshaya matematika [Tekst] V 3 t. T. 1: Elementy lineynoy algebry i analiticheskoy geometrii / Ya.S. Bugrov, S.M. Nikol'skiy. - M.: Drofa, 2004. 288 s.
12. Bugrov Ya.S. Vysshaya matematika [Tekst]. V 3 t. T. 2: Elementy lineynoy algebry i analiticheskoy geometrii: uchebnik dlya vuzov / Ya.S. Bugrov, S.M. Nikol'skiy. 7-e izd., ster. - M.: Izdatel'stvo Yurayt, 2023. - 281 s.
13. Vinogradov I.M. Matematicheskaya enciklopediya [Tekst]. V 5 t. T. 3: Koordinaty - Odnochlen / I.M. Vinogradov - M.: Sov. Enciklopediya, 1982. - 592 s.
14. Vinogradov I.M. Elementy vysshey matematiki. (Analiticheskaya geometriya. Differencial'noe ischislenie. Osnovy teorii chisel) [Tekst]: uchebnik dlya vuzov / I.M. Vinogradov. - M.: Vysshaya shkola, 1999. 511 s.
15. Voloshinov D.V. Algoritmicheskiy kompleks dlya resheniya zadach s kvadrikami s primeneniem mnimyh geometricheskih obrazov [Tekst] / D.V. Voloshinov // Geometriya i grafika. - 2020. - T. 8. - № 2. - S. 3-32. DOI:https://doi.org/10.12737/2308-4898-2020-3-3.
16. Girsh A.G. Novye zadachi nachertatel'noy geometrii. Prodolzhenie [Tekst] / A.G. Girsh // Geometriya i grafika. - 2022. - T. 10. - № 4. - S. 3-10. - DOI:https://doi.org/10.12737/2308-4898-2022-9-4-3-10.
17. Girsh A.G. Operaciya peresecheniya na kompleksnoy ploskosti [Tekst] / A.G. Girsh // Geometriya i grafika. 2021. - T. 9. - № 1. - S. 20-28. - DOI:https://doi.org/10.12737/23084898-2021-9-1-20-28.
18. Girsh A.G. Postroenie sfery po mnimym tochkam [Tekst] / A.G. Girsh // Geometriya i grafika. - 2022. T. 10. - № 3. - S. 3-11. - DOI:https://doi.org/10.12737/2308-48982022-10-3-3-11.
19. Zhiharev L.A. Otrazhenie ot krivolineynyh zerkal v ploskosti / L.A. Zhiharev // Geometriya i grafika. 2019. - T. 7. - № 1. - S. 46-54. - DOIhttps://doi.org/10.12737/article _5c9203adb22641.01479568.
20. Zhiharev L.A. Ploskie otrazheniya ot krivyh / L.A. Zhiharev, Yu.S. Karpova // Zhurnal estestvennonauchnyh issledovaniy. - 2020. - T. 5. - № 4. - S. 52-58.
21. Zinov'ev A.P. Modelirovanie opticheskih sistem v programmnom pakete "Resonator Studio" i ih eksperimental'naya realizaciya / A.P. Zinov'ev, M.V. Kol'cov, O.V. Martynova // Izvestiya vysshih uchebnyh zavedeniy. Radiofizika. - 2012. - T. 55. - № 12. S. 780-788.
22. Ivaschenko A.V. Obschiy analiz formy linii peresecheniya dvuh odnotipnyh poverhnostey vtorogo poryadka [Tekst] / A.V. Ivaschenko, D.A. Vavanov // Geometriya i grafika. - 2020. - T. 8. - № 4. - S. 24-34. - DOI:https://doi.org/10.12737/2308-4898-2021-8-4-24-34.
23. Ignat'ev S.A. Vizualizaciya zadach nachertatel'noy geometrii posredstvom Wolfram Mathematica [Tekst] / S.A. Ignat'ev, A.I. Folomkin, E.H. Muratbakeev // Geometriya i grafika. - 2020. - T. 8. - № 4. - S. 74-84. - DOI:https://doi.org/10.12737/2308-4898-2021-8-4-74-84.
24. Ignat'ev S.A. Funkcional'nye vozmozhnosti sredy Wolfram Mathematica dlya vizualizacii krivyh liniy i poverhnostey [Tekst] / S.A. Ignat'ev, A.I. Folomkin, E.H. Muratbakeev // Geometriya i grafika. - 2021. T. 9. - № 1. - S. 29-38. - DOI:https://doi.org/10.12737/2308-48982021-9-1-29-38.
25. Konopackiy E.V. Modelirovanie approksimiruyuschego 16-tochechnogo otseka poverhnosti otklika primenitel'no k resheniyu neodnorodnogo uravneniya teploprovodnosti [Tekst] / E.V. Konopackiy // Geometriya i grafika. - 2019. - T. 7. - № 2. - S. 39-46. - DOI:https://doi.org/10.12737/article_5d2c1a551a22c5.12136357.
26. Konopackiy E.V. Tochechnye instrumenty geometricheskogo modelirovaniya, invariantnye otnositel'no parallel'nogo proecirovaniya [Tekst] / E.V. Konopackiy, A.A. Bezditnyy // Geometriya i grafika. - 2022. T. 10. - № 4. - S. 11-21. - DOI:https://doi.org/10.12737/2308-48982022-9-4-11-21.
27. Korotkiy V.A. Approksimaciya fizicheskogo splayna s bol'shimi progibami [Tekst] / V.A. Korotkiy, I.G. Vitovtov // Geometriya i grafika. - 2021. - T. 10. № 1. - S. 3-19. - DOI:https://doi.org/10.12737/2308-4898-2021-9-13-19.
28. 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. - S. 19-30. - DOIhttps://doi.org/10.12737/22840
29. Korotkiy V.A. Konstruirovanie G2-gladkoy sostavnoy krivoy na osnove kubicheskih segmentov Bez'e [Tekst] / V.A. Korotkiy // Geometriya i grafika. 2021. - T. 9. - № 2. - S. 12-28. - DOI:https://doi.org/10.12737/23084898-2021-9-2-12-28.
30. Korotkiy V.A. Kubicheskie krivye v inzhenernoy geometrii [Tekst] / V.A. Korotkiy // Geometriya i grafika. - 2020. - T. 8. - № 3. - S. 3-24
31. Kulizina O.V. Primenenie ul'trazvuka dlya nerazrushayuschego kontrolya napryazheniy v sudostroitel'nyh stalyah [Tekst] / O.V. Kulizina, N.E. Nikitina // Nauchnye problemy vodnogo transporta. - 2010. - № 28. S. 164-168.
32. Leparov M.N. O geometrii, esche odin raz [Tekst] / M.N. Leparov // Geometriya i grafika. - 2022. - T. 10. № 1. - S. 3-13. - DOI:https://doi.org/10.12737/2308-4898-2022-101-3-13.
33. Michurov A.V. Raschet vliyaniya na akusticheskoe pole prelomleniy i otrazheniy na krivolineynyh poverhnostyah obolochek vrascheniya [Tekst] / A.V. Michurov, A.V. Sokolkin // Defektoskopiya. - 2020. - № 1. S. 31-43.
34. Muslimov E.R. Kompleksnoe ispol'zovanie metodov analiza svyazannyh voln i trassirovki luchey pri proektirovanii spektrografov s ob'emno-fazovymi difrakcionnymi reshetkami [Tekst] / E.R. Muslimov, N.K. Pavlycheva // HOLOEXPO 2019: Tezisy dokladov XVI mezhdunarodnoy konferencii po golografii i prikladnym opticheskim tehnologiyam, Sankt-Peterburg, 10-12 sentyabrya 2019 goda. - SPb.: Moskovskiy gosudarstvennyy tehnicheskiy universitet imeni N.E. Baumana (nacional'nyy issledovatel'skiy universitet), 2019. - S. 365-373.
35. Pritykin F.N. Metod zadaniya polozheniy uzlovyh tochek, opredelyayuschih konverty teney pri razlichnyh napravleniyah solnechnyh luchey i uglah povorota zdaniy [Tekst] / F.N. Pritykin, E.A. Kurysheva // Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya: Stroitel'stvo i arhitektura. - 2019. - T. 19. № 3. - S. 37-44.
36. Todorov N.F. Modelirovanie i issledovanie auralizacii pri rasprostranenii voln [Tekst]: avtoref. dis. … kand. tehn. nauk: 01.04.06 / N.F. Todorov. - Rostov n/D, 2014. - 162 s.
37. Sal'kov N.A. Ob izobrazheniyah [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2022. - T. 10. - № 2. S. 3-10. - DOI:https://doi.org/10.12737/2308-4898-2022-10-2-3-10.
38. Suncov O.S. Issledovanie otrazheniya ot krivolineynyh zerkal na ploskosti v programme Wolfram Mathematica / O.S. Suncov, L.A. Zhiharev // Geometriya i grafika. - 2021. - T. 9. - № 2. - S. 29-45. - DOIhttps://doi.org/10.12737/2308-4898-2021-9-2-29-45.
39. Sycheva A.A. Funkcional'no-voksel'noe modelirovanie krivyh Bez'e [Tekst] / A.A. Sycheva // Geometriya i grafika. - 2022. - T. 10. - № 4. - S. 63-72.
40. Tutova A.Yu. Ul'trazvuk v medicine [Tekst] / A.Yu. Tutova, M.G. Perepelkina // Vestnik nauki. - 2020. T. 3. - № 1. - S. 206-208.
41. Heyfec A.L. Vliyanie daty rascheta prodolzhitel'nosti insolyacii na parametry uplotnennoy zastroyki [Tekst] / A.L. Heyfec // Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya: Stroitel'stvo i arhitektura. - 2019. - T. 19. - № 3. - S. 61-70.
42. Bergmann U. High-resolution X-ray imaging based on curved Bragg mirrors: first results [Text] / U. Bergmann // IEEE Transactions on Nuclear Science. 2003. V. 50. I. 1, pp. 140-145.
43. Miguel A.L. Real-time 3D visualization of accurate specular reflections in curved mirrors a GPU implementation [Text] / A.L. Miguel, A.C. Nogueira, N. Goncalves // 2014 International Conference on Computer Graphics Theory and Applications (GRAPP). IEEE, 2014, pp. 1-8.
44. Minato A. Optical design of cube-corner retroreflectors having curved mirror surfaces [Text] / A. Minato, N. Sugimoto, Y. Sasano // Applied optics. 1992. V. 31. I. 28, pp. 6015-6020.
45. Ofek E. Interactive reflections on curved objects [Text] /E. Ofek, A. Rappoport // Proceedings of the 25 th annual conference on Computer graphics and interactive techniques. 1998, pp. 333-342.
46. Savarese S. Local shape from mirror reflections [Text] /S. Savarese, M. Chen, P. Perona // International Journal of Computer Vision. 2005. V. 64, pp. 31-67.
