Affine Focal Set - Singularity Theory Approach

Singularity Theory Approach

The idea here is to define a family of functions over M. The family will have the ambient real (n + 1)-space as its parameter space, i.e. for each choice of ambient point we will get a function defined over M. This family is the family of affine distance functions:

Given an ambient point and a surface point p, we can decompose the chord joining p to as a tangential component and a transverse component parallel to . The value of Δ is given implicitly in the equation

where Z is a tangent vector. We now seek the bifurcation set of the family Δ, i.e. the ambient points for which the restricted function

has degenertate 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 we differentiate the equation x - p = Z + ΔA. Let X be a tangent vector to M, and differentiate in that direction:

where I is the identity. This tells us that and . The last equality says that we have the following equation of differential one-forms . The Jacobian matrix will have zero determinant if, and only if, is degenerate as a one-form, i.e. for all tangent vectors X. Since it follows that is degenerate if, and only if, is degenerate. Since h is a non-degenerate two-form it follows that Z = 0. Notice that since M has a non-degenerate second fundamental form it follows that h is a non-degenerate two-form. 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 we consider the differential two-form . This is the two-form whose matrix representation is the Hessian matrix. We have already seen that we see that We have

.

Now assume that Δ has a singularity at p, i.e. Z = 0, then we have the two-form

.

We have also seen that, and so the two-form becomes

.

This is degenerate as a two-form if, and only if, there exists non-zero X for which it is zero for all Y. Since h is non-degenerate it must be that and . 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 where 1/t is an eigenvalue of S. The affine focal set!