Typical Wavefronts in 2D and 3D

This Demonstration shows all "generic" wavefronts and boundary wavefronts in the 2D plane and in 3D space. It also shows all generic intersections of such wavefronts. You can observe their shapes from any direction by rotating the 3D images, which will help your understanding of the structure of generic wavefronts.



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Wavefronts are defined by projected images of Legendrian submanifolds on Legendrian bundles. By Darboux's theorem, all Legendrian bundles are locally contact diffeomorphic to the canonical Legendrian bundle: for some , where is the 1-jet bundle of functions in variables with the contact structure defined by the canonical one-form . Thus, it is enough to consider the canonical Legendrian bundle for local situations. An -dimensional submanifold in is a Legendrian submanifold if vanishes on . The wavefront of is defined by . The theory of wavefronts describes stabilities of wavefronts generated by a hypersurface without boundary in a smooth manifold (see [2]).
The author constructs the theory of reticular Legendrian singularities that describes stabilities and genericities of wavefronts generated by a hypersurface with a boundary, a corner, or an -corner in a smooth manifold (see [3]). Let and for each
. Let represent a germ of the union of for all . We call a map germ a reticular Legendrian map if there exists a contact diffeomorphism germ on such that . The wavefront of is defined by the union of for all .
For the case , the hypersurface that has a boundary, the wavefront of is . The sets and correspond to the wavefronts generated by the hypersurface and its boundary, respectively. A smooth function germ defined on is a generating family of if is non-degenerate and
, .
Then the wavefronts of are given by
You can observe behaviors of wavefronts in a plane in Bifurcation of Boundary Wavefronts for Some Graphs.
Typical wavefronts are defined by generic reticular Legendrian maps and 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 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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