Geodesic
Quasi corner singularities
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 7-20

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We consider the classification of germs of functions up to a nonstandard equivalence relation similar to the quasi boundary equivalence and quasi equivalence of projections recently introduced by the second author. In fact, it is more rough than the classification of functions with respect to the group of diffeomorphisms preserving a corner (that is, a union of a pair of transversal hypersurfaces). We present the list of all simple classes and discuss its relation to the singularities of Lagrangian projections with corners. Also, we describe the bifurcation diagrams and caustics of simple quasi corner singularities. 
@article{TM_2010_270_a0,
     author = {F. Alharbi and V. Zakalyukin},
     title = {Quasi corner singularities},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {7--20},
     publisher = {mathdoc},
     volume = {270},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_270_a0/}
}
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F. Alharbi; V. Zakalyukin. Quasi corner singularities. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 7-20. http://geodesic.mathdoc.fr/item/TM_2010_270_a0/