Caustics, Wave Fronts, and Their Metamorphoses

  • Vladimir Igorevich Arnold


One of the most important deductions of singularity theory is the universality of certain simple forms like folds and cusps which one can expect to encounter everywhere and which it is useful to learn to recognise. As well as the previously enumerated singularities, one often meets some further types, which have been given their own names: the ‘swallowtail’, the ‘pyramid’ (which Thorn calls the ‘elliptic umbilic’), the ‘purse’ (which Thorn calls the ‘hyperbolic umbilic’), and so on.


Wave Front Catastrophe Theory Triaxial Ellipsoid Visible Contour Initial Front 
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Bibliographical Comments

For a different approach and different results from the theory developed in the other references, compare :

  1. G. Wassermann: Stability of unfoldings in space and time. Acta Math. 135 (1975), 57–128.MathSciNetzbMATHCrossRefGoogle Scholar

The picture of the metamorphoses of wave fronts first appeared in :

  1. V. I. Arnol’d: Critical points of smooth functions, in: Proceedings of the International Congress of Mathematicians (Vancouver, 1974) vol. 1. Canadian Mathematical Congress 1975, pp. 19–39.Google Scholar

The theory of metamorphoses of caustics and wave fronts depends on the results of:

  1. V. I. Arnol’d: Wave front evolution and equivariant Morse lemma. Commun. Pure Appl. Math. 29 (1976), 557–582.CrossRefGoogle Scholar

partially published in :

  1. V. M. Zakalyukin’s Thesis (in Russian). Moscow State University, 1978, 145 p.Google Scholar
  2. V. M. Zakalyukin: Reconstructions of wave fronts depending on one parameter. Funkts. Anal. Prilozh. 10:2 (1976), 69–70 (English translation: Funct. Anal. Appl. 10 (1976), 139-140).zbMATHGoogle Scholar
  3. V. M. Zakalyukin: Legendre mappings in Hamiltonian systems, in: Some problems of mechanics (in Russian). MAI, Moscow 1977, pp. 11–16.Google Scholar

The drawing of the metamorphoses of caustics was first published in the first Russian edition of this pamphlet:

  1. V. I. Arnol’d: Catastrophe theory (in Russian). Priroda, Issue 10, 1979, 54–63.Google Scholar
  2. In the French translation of this article by J. M. Kantor (Matematica, May 1980, 3-20), this picture was replaced by a page of R. Thorn’s comments.Google Scholar
  3. V. M. Zakalyukin: Reconstructions of fronts and caustics depending on a parameter and versality of mappings, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 22, Viniti, Moscow 1983, pp. 56–93 (English translation: J. Sov. Math. 27 (1984), 2713-2735).Google Scholar

For the theory of bicaustics, see :

  1. V. I. Arnol’d: Restructurings of singularities of potential flows in a collision-free medium and metamorphoses of caustics in three-dimensional space (in Russian). Tr. Semin. Im. I. G. Petrovskogo 8 (1982), 21–57.Google Scholar
  2. These results were announced at the Petrovskij seminar, autumn 1980 (see Usp. Mat. Nauk 36:3 (1981) and the pictures first appeared in the 1981 Russian edition of this book. Some of these surfaces appear later in the work of Shcherbak and of Gaffney and du Plessis (1982) in other situations (as unions of space curve tangents in Shcherbak’s version).Google Scholar

The classification of singularities of caustics and wave fronts up to dimension 10:

  1. V. M. Zakalyukin: Lagrangian and Legendrian singularities. Funkts. Anal. Prilozh. 10:1 (1976), 26–36, (English translation: Funct. Anal. Appl. 10 (1976), 23-31).Google Scholar

and corrections:

  1. V. I. Arnol’d, S. M. Gusein-Zade, A. N. Varchenko: Singularities of differentiable maps I: The classification of critical points, caustics and wave fronts. Nauka, Moscow 1982, § 21 (English translation: Birkhäuser, Boston 1985)Google Scholar

The work on ice motion:

  1. J. F. Nye, A. S. Thorndike: Events in evolving three-dimensional vector fields. J. Phys. A 13 (1980), 1–14.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Vladimir Igorevich Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

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