This means that the derivative of an affine principal

*curvature*in its own affine principal direction is zero. 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 arise. The affine focal set of a plane curve will generically consist of smooth pieces of curve and ordinary cusp points (semi-cubical parabolae).