If we consider a plane curve that is a Legendrian front with a singularity of type A2, we say that this singularity is a cusp. If we consider a surface of (for instance) R^3 that is a Legendrian front with a singularity of (transversal) type A2, we say that this singularity is a cuspidal edge. Is there an appropriate terminology in higher dimension(s). For instance, what could we say for a hypersurface of R^4 that is a Legendrian front with a singularity of (transversal type) A2? I would like something understandable by a non-specialist.
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