Acta Applicandae Mathematicae

, Volume 137, Issue 1, pp 17–39 | Cite as

Near-Horizon Celestial Phenomena, a Study in Geometric Optics

  • Henk W. BroerEmail author


Near horizon phenomena, like blank strips in the setting sun or Fata Morganas, can be largely understood in terms of geometric optics. Here the existence of a warm layer above the observer in which light rays can be either refracted of reflected, already suffices to explain many of the phenomena; a reasoning which goes back to Alfred Wegener in the 1920’s. We review this theory, where first the atmosphere is divided into discrete, optically homogeneous layers, and where later the atmosphere is considered as a continuous, isotropic medium. Thus the theory gets embedded in differential geometry, where light rays are geodesics. A simplifying assumption is that the refraction index only depends on the elevation above the surface of the earth. Here a connection is made with the geodesic problem on a surface of revolution. In the background of this we touch the variational calculus as this runs from Bernoulli to Hamilton, all the way maintaining a completely anachronistic viewpoint.


Geometric optics Differential geometry Nonlinear dynamics 


  1. 1.
    Andersen, K.: The mathematical technique in Fermat’s deduction of the law of refraction. Hist. Math. 10, 48–62 (1983)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. GTM, vol. 60. Springer, Berlin (1978). Second edition. Springer, Berlin (1989)zbMATHGoogle Scholar
  3. 3.
    Arnold, V.I.: Catastrophe Theory. Springer, Berlin (1984)zbMATHGoogle Scholar
  4. 4.
    Arnold, V.I.: Singularities of Caustics and Wavefronts. Mathematics and Its Applications (Soviet Series), vol. 62. Kluwer Academic, Norwell (1990)CrossRefGoogle Scholar
  5. 5.
    Arnold, V.I.: On the topological properties of Legendrian projections in the contact geometry of wave fronts. Algebra Anal. 6(3), 1–16 (1994)Google Scholar
  6. 6.
    Bernoulli, J.: In: Cramer, G. (ed.) Opera Johannis Bernoullii, vol. 4 (1742). GenéveGoogle Scholar
  7. 7.
    Berry, M.V., Upstill, C.: Catastrophe optics: morphologies of caustics and their diffraction patterns. Prog. Opt. 18, 257–346 (1980)CrossRefGoogle Scholar
  8. 8.
    Broer, H.W.: Hemelverschijnselen nabij de Horizon, naar Minnaert en Wegener, Bernoulli en Hamilton. Epsilon Uitgaven (2013); Near-horizon celestial phenomena: inspired by Minnaert and Wegener, Bernoulli and Hamilton. MAA. Carus series (to appear)Google Scholar
  9. 9.
    Broer, H.W.: Bernoulli’s lichtstraal-oplossing van het brachistochrone probleem door de ogen van Hamilton. Nieuw Archief voor Wiskunde, 5th series, 14(2), 98–107 (2013); Bernoulli’s light ray solution of the brachistochrone problem through Hamilton’s eyes. Int. J. Bif. Chaos (2014) (to appear)Google Scholar
  10. 10.
    Carathéodory, C.: Gesammelte Mathematische Schriften, vols. I–IV. Beck, München (1954–1957). I–VGoogle Scholar
  11. 11.
    Dijksterhuis, E.J.: The Mechanizaton of the World Picture: Pythagoras to Newton. Princeton University Press, Princeton (1986). De Mechanisering van het Wereldbeeld. Meulenhoff (1950). Third edition (1977)Google Scholar
  12. 12.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall International, Englewood Cliffs (1976)zbMATHGoogle Scholar
  13. 13.
    Euler, L.: Leonhardi Euleri Opera Omnia (1911–1975). vols. 72, BernGoogle Scholar
  14. 14.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vols. 1–3. Addison-Wesley, Reading (1963–1965)Google Scholar
  15. 15.
    Floor, C.: De ondergaande zon. Zenit 4, 126–128 (1978)Google Scholar
  16. 16.
    Floor, C.: Atmosferische straalkromming. In: Natuur en Techniek (1978)Google Scholar
  17. 17.
    Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1950). Second edition (1980)Google Scholar
  18. 18.
    Goldstine, H.H.: A History of the Calculus of Variations from the 17th Through the 19th Century. Studies in the History of Mathematics and Physical Sciences, vol. 5. Springer, Berlin (1980)zbMATHGoogle Scholar
  19. 19.
    Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys, vol. 14. Am. Math. Soc., Providence (1977)zbMATHGoogle Scholar
  20. 20.
    Hamilton, W.R.: Theory of systems of rays. Trans. R. Ir. Acad. 15, 69–174 (1828)Google Scholar
  21. 21.
    Hamilton, W.R.: On a general method in dynamics. Philos. Trans. R. Soc. 124, 247–308 (1834)CrossRefGoogle Scholar
  22. 22.
    Hamilton, W.R.: Second essay on a general method in dynamics. Philos. Trans. R. Soc. 125, 95–144 (1835)CrossRefGoogle Scholar
  23. 23.
    Humphries, W.J.: Physics of the Air. Franklin Institute, Philadelphia (1920). Third edition. McGraw-Hill (1940); Dover (1965)Google Scholar
  24. 24.
    Huygens, C.: Traité de la Lumière. In: Œvres Complètes de Christiaan Huygens publiées par la Société Hollandaise des Sciences. Martinus Nijhoff, The Hague (1929). Treatise on Light. Dover (1962); Verhandeling over het Licht. Epsilon Uitgaven 18 (1990)Google Scholar
  25. 25.
    Kepler, J.: In: Ad Vitellium Paralipomena, quibus Astronimiae pars optica traditur. Claude Marne and Erven Johannes Auber. Frankfurt (1604); In: von Dyck, W., Caspar, M. (eds.) Gesammelte Werke, band II. C.H. Becksche Verlagsbuchsabhandlungen. München (1938)Google Scholar
  26. 26.
    Lagrange, J.L.: In: Serret, A., Darboux, G. (eds.) Œuvres (1867–1892). ParisGoogle Scholar
  27. 27.
    Lehn, W.H.: Inversion of superior mirage data to compute temperature profiles. J. Opt. Soc. Am. 73(12), 1622–1625 (1983)CrossRefGoogle Scholar
  28. 28.
    Leibniz, G.W.: Nova methodus pro maximis et minimis. Acta Eruditorum (1684). In: Struik, D.J. (ed.), A Source Book in Mathematics, vols. 1200–1800, pp. 271–281. Harvard University Press, Cambridge (1969)Google Scholar
  29. 29.
    Leonhardt, U., Philbin, Th.: Geometry and Light: The Science of Invisibility. Dover, Dover (2010)Google Scholar
  30. 30.
    Levi, M.: The Mathematical Mechanic Using Physical Reasoning to Solve Problems. Princeton University Press, Princeton (2009)CrossRefzbMATHGoogle Scholar
  31. 31.
    Minnaert, M.G.J.: The Nature of Light & Colour in the Open Air. Dover (1954); De Natuurkunde van ’t Vrije Veld, delen 1, 2 en 3, Vijfde Editie, ThiemeMeulenhoff (1996). First edition (1937–1940)Google Scholar
  32. 32.
    Noether, E.: Invariante Variationsprobleme. In: Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math-Phys. Klasse, pp. 235–257 (1918)Google Scholar
  33. 33.
    O’Connell, D.J.K.: The Green Flash and Other Low Sun Phenomena. North-Holland, North-Holland (1958)Google Scholar
  34. 34.
    Pierce, A.D.: Acoustics: An Introduction to Its Physical Principles and Applications. McGraw-Hill, New York (1981). Also available from the Acoustical Society of AmericaGoogle Scholar
  35. 35.
    Poston, T., Stewart, I.: Catastrophe Theory and Its Applications. Pitman, London (1978)zbMATHGoogle Scholar
  36. 36.
    Riemann, G.F.B.: Über die Hypothesen, welche der Geometrie zu Grunde liegen. Habilitationsschrift (1854). Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868)Google Scholar
  37. 37.
    Sabra, A.I.: Theories of Light from Descartes to Newton. Osbourne, London (1967)Google Scholar
  38. 38.
    Spivak, M.: Differential Geometry, vol. I. Publish or Perish, Houston (1970)Google Scholar
  39. 39.
    van der Werf, S.Y.: Het Nova Zembla Verschijnsel; geschiedenis van een luchtspiegeling. In: Historische Uitgeverij (2011)Google Scholar
  40. 40.
    van der Werf, S.Y., Können, G.P., Lehn, W.H., Steenhuizen, F.: Waerachtige beschrijvingen van het Nova Zembla effect. Ned. Tijdschr. Natuurkd. 66, 120–126 (2000)Google Scholar
  41. 41.
    van Maanen, J.A.: Een Complexe Grootheid, leven en werk van Johann Bernoulli, 1667–1748. vol. 34. Epsilon Uitgaven, Amsterdam (1995)Google Scholar
  42. 42.
    Verne, J.: Le Rayon Vert. Hetzel, Paris (1881)Google Scholar
  43. 43.
    Wegener, A.L.: Über die Ursache der Zerrbilder bei Sonnenuntergängen. Beitr. Phys. Freien Atmos. Strasbourg 4, 26–34 (1912)Google Scholar
  44. 44.
    Wegener, A.L.: Optik der Atmosphäre. In: Müller, Pouillet (eds.) Lehrbuch der Physik, 2nd edn., pp. 266–289. Viewig & Son, Braunschweig (1928)Google Scholar

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.GroningenThe Netherlands

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