Typical Bifurcations of Wavefronts in 2D and 3D

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This Demonstration shows all "generic" bifurcations of boundary wavefronts in 2D and 3D. You can control the time and view point. Check "initialize" to change the type of wavefronts.

Contributed by: Takaharu Tsukada (September 2010)
Open content licensed under CC BY-NC-SA



The author constructs the theory of reticular Legendrian unfoldings that describes stable and generic bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an r‐corner (cf. [4]).

In the case , the hypersurface has no boundary; this is known as the theory of perestroikas of fronts ([1], the figures are given on p. 60). A one-parameter family of wavefronts is given by a generating family defined on such that


In the case , the hypersurface has a boundary; a reticular Legendrian unfolding gives the wavefront , where the set is the wavefront generated by the hypersurface at time and the set is the wavefront generated by the boundary of the hypersurface at time .

A reticular Legendrian unfolding has a generating family. Then the wavefront is given by the generating family defined on such that


Typical bifurcations of wavefronts in 2D and 3D are defined by generic reticular Legendrian unfoldings for the cases . Their generating families are stably reticular ‐equivalent to one of the following.

In the case :

, ,

In the case :

The wavefronts for singularities are given in the Demonstration Typical Wavefronts in 2D and 3D.


[1] V. I. Arnold, Singularities of Caustics and Wave Fronts, Dordrecht: Kluwer Academic Publishers, 1990.

[2] V. I. Arnold, S. M. Gusein–Zade, and A. N. Varchenko, Singularities of Differential Maps I, Basel: Birkhäuser, 1985.

[3] T. Tsukada, "Genericity of Caustics and Wavefronts on an r-Corner", 14(3), The Asian Journal of Mathematics, 2010 pp. 335–358.

[4] T. Tsukada, "Bifurcations of Wavefronts on r-Corners: Semi-Local Classification," 18(3) Methods and Applications of Analysis, 2011 pp. 303–334.

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