Affine Focal Set

In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold M embedded in a smooth manifold N is the caustic generated by the affine normal lines. It can be realised as the bifurcation set of a certain family of functions. The bifurcation set is the set of parameter values of the family which yield functions with degenerate singularities. This is not the same as the bifurcation diagram in dynamical systems.

Let us assume that M is an n-dimensional smooth hypersurface in real (n+1)-space. We assume that M has no points where the second fundamental form is degenerate. We recall from the article affine differential geometry that there is a unique transverse vector field over M. This is the affine normal vector field, or the Blaschke normal field. A special (i.e. det = 1) affine transformation of real (n + 1)-space will carry the affine normal vector field of M onto the affine normal vector field of the image of M under the transformation.