Geodesic
Polyhedral and Dihedral Caustics in the~$\mathbb R^3$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 195-201

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The role of the symmetries in the topology of sets of Lagrangian singularities is studied in a simple physical model: the envelope of the rays emanating from a convex wave front invariant under the action of discrete subgroups of $O(3)$. New point-singularities of integer index are found. They are located at the vertices of the polyhedron or of its dual. For the dihedral subgroups, we have found a remarkable property of stability of umbilics. These properties result from the interplay between the symmetries of the singularities and the topology of the wave front. An application to fine-particle magnetic systems is given. 
@article{TM_1999_225_a12,
     author = {A. Joets and M. I. Monastyrskii and R. Ribotta},
     title = {Polyhedral and {Dihedral} {Caustics} in the~$\mathbb R^3$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {195--201},
     publisher = {mathdoc},
     volume = {225},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_225_a12/}
}
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A. Joets; M. I. Monastyrskii; R. Ribotta. Polyhedral and Dihedral Caustics in the~$\mathbb R^3$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 195-201. http://geodesic.mathdoc.fr/item/TM_1999_225_a12/