Multi-Time Bifurcations of Wavefronts in 2D and 3D

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This Demonstration shows all "generic" multi-bifurcations of wavefronts in 2D and 3D. You can control the time.

Contributed by: Takaharu Tsukada (October 2013)
Open content licensed under CC BY-NC-SA



In [5], the author constructs the theory of reticular Legendrian unfoldings that describes stable and generic multi-bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an ‐corner.

In the case , the hypersurface has no boundary; a two-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 multi-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 :

, , , .


[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, Volume I: The Classification of Critical Points, Caustics and Wave Fronts (Monographs in Mathematics, Vol. 82), Basel: Birkhäuser, 1985.

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

[4] T. Tsukada, "Bifurcations of Wavefronts on -Corners: Semi-Local Classification," Methods and Applications of Analysis, 18(3), 2011 pp. 303–334. doi:10.4310/MAA.2011.v18.n3.a3.

[5] T. Tsukada. "Multi-Bifurcations of Wavefronts on -Corners."

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