## Return to Answer

2 added 747 characters in body

I've been trying to work myself through this very problem of late. A text you might find useful is Geometric Differentiation by Ian Porteous. Therein the author through his own idiosyncratic yet quite utile methodology discusses the various level of contact that spheres can have with a given surface and hence involving the so called ridge points.

As alluded to in previous posters, I think that one of the key differences between the two loci is that of local vs global phenomena on a surface. Focal loci are local in nature, whereas cut loci are global.

Edit: This paper -- J.J. Hebda, Cut Loci of Submanifolds in Space Forms and in the Geometries of Moebius and Lie, Geometriae Dedicata, 55, 75-93, 1995 -- gives the following characterization of cut loci of submanifolds (which he takes to mean the entire set of focal points of the given submanifold and the points with two or more closest points on the submanifold): The set of cut points of a properly embedded submanifold of a complete, connected space form (i.e., Euclidean space, hyperbolic space or the sphere) is the set of centers of maximal supporting balls of that submanifold in the larger space, where a supporting ball is an open ball in the space-form that does not intersect the submanifold but whose boundary sphere does.

1

I've been trying to work myself through this very problem of late. A text you might find useful is Geometric Differentiation by Ian Porteous. Therein the author through his own idiosyncratic yet quite utile methodology discusses the various level of contact that spheres can have with a given surface and hence involving the so called ridge points.

As alluded to in previous posters, I think that one of the key differences between the two loci is that of local vs global phenomena on a surface. Focal loci are local in nature, whereas cut loci are global.