한국수학사학회지 (Journal for History of Mathematics)

한국수학사학회 (The Korean Society for History of Mathematics)
권호 표지
DOI QR코드

DOI QR Code

포락선 연구의 역사와 포락선의 여러 가지 정의

History of Studies on the Envelope Curves and Various Definitions of the Envelope

  • 투고 : 2024.05.09
  • 심사 : 2024.08.25
  • 발행 : 2024.08.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Research on the caustic, the envelope of light rays, began with the geometric optics studies of Huygens and others in the 17th century. One of the most important problems in optics in the 17th century was focusing the sun's rays. This was a problem that had to be solved in order to manufacture various practical optical instruments at the time. Beginning with research on the caustic during this period, the concept of envelope became generalized and expanded to various fields until the 19th century. This paper examines the mathematical history involved in the study of envelope curves. We compare several methods of defining the envelope and provide an example of calculating the envelope accordingly.

키워드

참고문헌

  1. V. ARNOLD, Mathematical Methods of Classical Mechanics, translated from the Russian by K. Vogtmann and A. Weinstein, second edition, Graduate Texts in Mathematics 60, Springer-Verlag, New York, 1980. https://doi.org/10.1007/978-1-4757-2063-1
  2. V. ARNOLD, Singularities of caustics and wave fronts, Mathematics and its Applications (Soviet Series) 62, Kluwer Academic Publishers Group, Dordrecht, 1990. https://doi.org/10.1007/978-94-011-3330-2
  3. V. ARNOLD, et al., Singularity Theory. II, Dynamical Systems VIII, Encyclopaedia Math. Sci., Vol. 39, Springer-Verlag, Berlin, 1993.
  4. V. ARNOLD, S. GUSEIN-ZADE, A. VARCHENKO, Singularities of differentiable maps. Vol. I, translated from Russian by Ian Porteous and Mark Reynolds, Monographs in Mathematics 82, Birkhauser Boston, Inc., Boston, MA, 1985. https://doi.org/10.1007/978-1-4612-5154-5
  5. T. BARANIECKI, et al., System for laser microcladding of metal powders, Przeglad Spawalnictwa 9, 2011.
  6. F. BERON-VERA, et al., Ray dynamics in a long-range acoustic propagation experiment, J. Acoust. Soc. Am. 114 (2003), 1226-1242. https://doi.org/10.1121/1.1600724
  7. J. BRUCE, P. GIBLIN, What Is an Envelope?, The Mathematical Gazette 65(433) (1981), 186-192. https://doi.org/10.2307/3617131
  8. A. CAYLEY, A memoir upon caustics, Philos. Trans. Roy. Soc. London 147 (1857), 273-312; Collected Works 2. 336-380. https://doi.org/10.1098/rstl.1857.0014
  9. A. CLAUSEN, C. STRUB, A general and intuitive envelope thorem, Edinburgh School of Economics Discussion Paper Series, ESE Discussion Papers, No. 274, 2016.
  10. J. DARBOUX, Lecons sur la Theorie Generale des Surfaces, Band 2, Buch 5, 1894.
  11. M. FRIEDMAN, K. KRAUTHAUSEN, (edited) Model and Mathematics: From the 19th to the 21st Century, 2022, 180-181. https://doi.org/10.1007/978-3-030-97833-4
  12. R. HEATH, Geometrical Optics, Cambridge University Press, 1887.
  13. R. HERMAN, A Treatise on Geometrical Optics, Cambridge University Press, 1900.
  14. C. HUYGENS, Abhandlung uber das licht (written 1678, published 1690), German transl., Leipzig, 1913.
  15. A. KNESER, Ableitung hinreichender Bedingungen des Maximum oder Minimum einfacher Integrale aus der Theorie der Zweiten Variation, Math.Annalen, Band 51, 1898.
  16. J. LAWRENCE, A Catalog of Special Plane Curves, Dover, New York, 1972.
  17. E. LOCKWOOD, A Book of Curves, Cambridge University Press, 1961.
  18. B. MCDONALD, W. KUPERMAN, Time domain formulation for pulse propagation including nonlinear behavior at a caustic, J. Acoust. Soc. Am., 81 (1987), 1406-1417. https://doi.org/10.1121/1.394546
  19. P. MILGROM, I. SEGAL, Envelope theorems for arbitrary choice sets, Econometrica, 70(2) (2002), 583-601. https://doi.org/10.1111/1468-0262.00296
  20. S. PATELA, Principle of operation, properties and parameters of optical fibers, 2013. http://www-old.wemif.pwr.wroc.pl/spatela/pdfy/0020.pdf
  21. B. PERCEL, The effect of caustics in acoustics. inverse scattering experiments, PhD dissertation, Houston, Texas: Rice University; 1989. http://www.caam.rice.edu/caam/trs/89/TR89-03.pdf
  22. V. POHL, How to Enrich Geometry Using String Designs, National Council of Teachers of Mathematics, 1986.
  23. T. ROW, Geometric Exercises in Paer Folding, W. W. Beman and E. E. Smith, eds, Open Court, 1917.
  24. G. SCARPELLO, A. SCIMONE, The Work of Tschirnhaus, La Hire and Leibniz on Catacaustics and the Birth of Envelopes of Lines in the 17th Century, Arch. Hist. Exact Sci. 59 (2005), 223-250. https://doi.org/10.1007/s00407-004-0092-7
  25. P. von SEIDEL, Uber die Brennflache eines Strahlenbundels, welches durch ein System von centrirten spharischen Glasern hindurch gegangen ist, Monatsberichte der Koniglichen Preussische Akademie der Wissenschaften zu Berlin, (1862) 695-705.
  26. D. STRUIK, Outline of a History of Differential Geometry: I, Isis, Apr., 1933, 19(1) (1933), 92-120. https://doi.org/10.1086/346723
  27. R. THOM, Sur la theorie des enveloppes, J. Math. Pures Appl. 41(9) (1962), 177-192.
  28. J. VINER, Cost Curves and Supply Curves, Zeitschrift fur Nationalokonomie 3(1) (1931), 23-46. https://doi.org/10.1007/BF01316299
  29. R. YATES, Curves and Their Properties, National Council of Teachers of Mathematics, 1952 (reprinted 1974).
  30. E. ZERMELO, Untersuchungen zur Variations-Rechnung, Berlin(dissertation), 1894.