Chelyabinsk, Chelyabinsk, Russian Federation
Chelyabinsk, Chelyabinsk, Russian Federation
Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.
affine compression; cubic curve; Ferguson curve; polynomial model; parametric model; vector derivative; graphic differentiation
1. Voloshinov D.V. Algoritmicheskiy kompleks dlya resheniya zadach s kvadrikami s primeneniem mnimyh geometricheskih obrazov / D.V. Voloshinov // Geometriya i grafika. - 2020. - T. 8. - № 2. - S. 3-32. - DOIhttps://doi.org/10.12737/2308-4898-2020-3-32.
2. Voloshinov D. V. Konstruktivnoe geometricheskoe modelirovanie. Teoriya, praktika, avtomatizaciya: monografiya [Tekst] / D.V Voloshinov. - Saarbrucken: Lambert Academic Publishing, 2010. - 355 s.
3. Voloshinov D.V. Konstruktivnoe geometricheskoe modelirovanie kak perspektiva prepodavaniya graficheskih disciplin [Tekst] / D.V. Voloshinov, K.N. Solomonov // Geometriya i grafika. - 2013. - T. 1. - № 2. - S. 10-13. - DOIhttps://doi.org/10.12737/778.
4. Golovanov N.N. Geometricheskoe modelirovanie / N.N. Golovanov. - M.: Izd-vo fiziko-matematicheskoy literatury, 2012. - 472 s.
5. Zav'yalov Yu.S. Splayny v inzhenernoy geometrii / Yu.S. Zav'yalov, V.A. Leus, V.A. Skorospelov. - M.: Mashinostroenie, 1985. - 224 s.
6. Ivanov G.S. Teoreticheskie osnovy nachertatel'noy geometrii / G.S. Ivanov. - M.: Mashinostroenie, 1998. - 157 s.
7. Kurs nachertatel'noy geometrii (s uchetom principov programmirovannogo obucheniya) / pod red. N.F. Chetveruhina. - M.: Vysshaya shkola, 1968. - 266 s.
8. Konopackiy E.V. Vychislitel'nye algoritmy modelirovaniya odnomernyh obvodov cherez k napered zadannyh tochek / E.V. Konopackiy, A.A. Krys'ko, A.I. Bumaga // Geometriya i grafika. - 2018. - T. 6. - № 3. - S. 20-32. - DOI:https://doi.org/10.12737/article_5bc457ece18491.72807735.
9. Korotkiy V.A. Kubicheskie krivye v inzhenernoy geometrii / V.A. Korotkiy // Geometriya i grafika. - 2020. - T. 8. - № 3. - S. 3-24. - DOI:https://doi.org/10.12737/2308-4898-2020-3-24.
10. Lyubchinov E. V. O gladkosti stykovki liniy i poverhnostey pri ciklograficheskom modelirovanii poverhnostnyh form avtomobil'nyh dorog / E.V. Lyubchinov, K.L. Panchuk // Vestnik YuUrGU. Seriya “Stroitel'stvo i arhitektura”. - 2020. - T. 20. - № 1. - S. 52-62. - DOI:https://doi.org/10.14529/build200106.
11. Nazarova O.N. Sovremennye problemy prepodavaniya kursa “Prikladnaya geometriya i inzhenernaya grafika” dlya ekspluatacionnyh napravleniy aviacionnogo vuza / O.N.Nazarova // Geometriya i grafika. - 2020. - T. 8. - № 2. - S. 58-65. - DOI:https://doi.org/10.12737/2308-4898-2020-58-65.
12. Pontryagin L.S. Kubicheskaya parabola / L.S. Pontryagin // Nauchno-populyarnyy fiziko-matematicheskiy zhurnal «Kvant». - 1984. - №3. - S. 10-14, 32.
13. Popov E.P. Nelineynye zadachi statiki tonkih sterzhney / E.P. Popov. - M.: GITTL, 1948. - 172 s.
14. Prasolov V.V. Geometriya / V.V. Prasolov, V.M. Tihomirov. - M.: Izd-vo MCNMO, 2013. - 336 s.
15. Preparata F. Vychislitel'naya geometriya / F. Preparata, M. Sheymos. - M.: Mir, 1989. - 478 s.
16. Ryazanov S.A. Raschet koordinat modificirovannogo profilya proizvodyaschey poverhnosti zuboreznogo instrumenta / S.A. Ryazanov, M.K. Reshetnikov // Geometriya i grafika. - 2020. - T. 8. - № 4. - S. 35-46. - DOI:https://doi.org/10.12737/2308-4898-2021-8-4-35-46.
17. Savelov A.A. Ploskie krivye / A.A. Savelov. - M.: Knizhnyy dom «Librokom», 2009. - 296 s.
18. Savel'ev Yu.A. Vychislitel'naya grafika v reshenii netradicionnyh inzhenernyh zadach / Yu.A. Savel'ev, E.Yu. Cherkasova // Geometriya i grafika. - 2020. - T. 8. - № 1. - S. 33-44. - DOI:https://doi.org/10.12737/2308-4898-2020-33-44.
19. Sal'kov N.A. Geometricheskaya sostavlyayuschaya tehnicheskih innovaciy / N.A. Sal'kov // Geometriya i grafika. - 2018. - T. 6. - № 2. - S. 85-93. - DOI:https://doi.org/10.12737/article_5b55a5163fa053.07622109.
20. Sal'kov N.A. Kachestvo geometricheskogo obrazovaniya pri razlichnyh podhodah k metodike obucheniya / N.A. Sal'kov // Geometriya i grafika. - 2020. - T. 8. - № 4. - S. 47-60. - DOI:https://doi.org/10.12737/2308-4898-2021-8-4-47-60.
21. Sal'kov N.A. Fenomen prisutstviya nachertatel'noy geometrii v drugih uchebnyh disciplinah / N.A. Sal'kov, N.S. Kadykova // Geometriya i grafika. - 2020. - T. 8. - № 4. - S. 61-73. - DOI:https://doi.org/10.12737/2308-4898-2021-8-4-61-73.
22. Suhih B.I. Vychislitel'naya geometriya. Osnovnye ob'ekty i preobrazovaniya: uchebnoe posobie / B.I. Suhih, R.A. Vaysburd. - Ekaterinburg, izd-vo UPI, 1989. - 92 s.
23. Usataya T.V. Sovremennye podhody k proektirovaniyu izdeliy v processe obucheniya studentov komp'yuternoy grafike / T.V. Usataya, L.V. Deryabina, E.S. Reshetnikova // Geometriya i grafika. - 2019. - T. 7. - № 1. - S. 74-82. - DOIhttps://doi.org/10.12737/article_5c91fd2bde0ff7.07282102.
24. Uoker, R. Algebraicheskie krivye / R. Uoker. - M.: Knizhnyy dom «Librokom», 2009. - 240 s.
25. Foks A. Vychislitel'naya geometriya. Primenenie v proektirovanii i na proizvodstve / A. Foks, M. Pratt. - M., Mir, 1982. - 304 s.
26. Shikin E.V. Krivye i poverhnosti na ekrane komp'yutera / E.V. Shikin, L.I. Pliss. - Dialog-MIFI, 1996. - 240 s.
27. Glaeser, G. Geometrie und ihre Anwendungen in Kunst, Natur und Technik / G. Glaeser. - Springer Spektrum, 2014. - 508 pp. DOIhttps://doi.org/10.1007/978-3-642-41852-5.
