It can be realised as the bifurcation set of a certain family of functions.
Assume that M is an n-dimensional smooth hypersurface in real (n+1)-space.
Assume that M has no points where the second fundamental form is degenerate.
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.
The affine normal vector field will be denoted by
At each point of M it is transverse to the tangent space of M, i.e. For a fixed
may be parametrised by t where The affine focal set is given geometrically as the infinitesimal intersections of the n-parameter family of affine normal lines.
To calculate, choose an affine normal line, say at point p; then look at the affine normal lines at points infinitesimally close to p and see if any intersect the one at p. If p is infinitesimally close to
This can be done by using power series expansions, and is not too difficult; it is lengthy and has thus been omitted.
, where D is the covariant derivative on real (n + 1)-space (for those well read: it is the usual flat and torsion free connexion).
denotes the greatest integer function, there will generically be (n − 2k)-pieces of the affine focal set above each point p. The −2k corresponds to pairs of eigenvalues becoming complex (like the solution to
The affine focal set need not be made up of smooth hypersurfaces.
In fact, for a generic hypersurface M, the affine focal set will have singularities.
and a surface point p, it is possible to decompose the chord joining p to
The value of Δ is given implicitly in the equation where Z is a tangent vector.
Now, what is sought is the bifurcation set of the family Δ, i.e. the ambient points for which the restricted function has degenerate singularity at some p. A function has degenerate singularity if both the Jacobian matrix of first order partial derivatives and the Hessian matrix of second order partial derivatives have zero determinant.
To discover if the Jacobian matrix has zero determinant, differentiating the equation x - p = Z + ΔA is needed.
Let X be a tangent vector to M, and differentiate in that direction: where I is the identity.
Since Z = 0 the set of ambient points x for which the restricted function
has a singularity at some p is the affine normal line to M at p. To compute the Hessian matrix, consider the differential two-form
What remains is Now assume that Δ has a singularity at p, i.e. Z = 0, then we have the two-form It also has been seen that
So the singularity is degenerate if, and only if, the ambient point x lies on the affine normal line to p and the reciprocal of its distance from p is an eigenvalue of S, i.e. points
The affine focal set can be the following: To find the singular points, simply differentiate p + tA in some tangent direction X: The affine focal set is singular if, and only if, there exists non-zero X such that
Standard ideas can be used in singularity theory to classify, up to local diffeomorphism, the affine focal set.
The family of affine distance functions should be a versal unfolding of the singularities which arise.
The affine focal set of a plane curve will generically consist of smooth pieces of curve and ordinary cusp points (semi-cubical parabolae).
The affine focal set of a surface in three-space will generically consist of smooth pieces of surface, cuspidal cylinder points (
The question of the local structure in much higher dimension is of great interest.
For example, it is possible to construct a discrete list of singularity types (up to local diffeomorphism).
In much higher dimensions, no such discrete list can be constructed, as there are functional moduli.