Affine Focal Set - Singular Points

Singular Points

The affine focal set can be the following:

To find the singular points we 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, i.e. if, and only if, X is an eigenvector of S and the derivative of t in that direction is zero. This means that the derivative of an affine principal curvature in its own affine principal direction is zero.