Local Structure
We can use the standard ideas in singularity theory to classify, up to local diffeomorphism, the affine focal set. If the family of affine distance functions can be shown to be a certain kind of family then the local structure is known. We want the family of affine distance functions to be a versal unfolding of the singularities which arrise.
The affine focal set of a plane curve will generically consist of smooth pieces of curve and ordinary cusp points (semi-cubical palabara|semi-cubical parabolae).
The affine focal set of a surface in three-space will generically consist of smooth pieces of surface, cuspidal cylinder points, swallowtail points, purse points, and pyramid points . The and series are as in Arnold's list.
The question of the local structure in much higher dimension is of great interest. For example, we were able to construct a discrete list of singularity types (up to local diffeomprhism). In much higher dimensions no such discrete list can be constructed, there are functional modulii.
Read more about this topic: Affine Focal Set
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