dg.differential geometry – Calculating a nested manifold

Given a “convex” manifold $A$, how can the nested manifold $B$ be calculated whose tangent lines intersect $A$ in exactly two points $p$ and $q$ at a fixed positive distance $|q-p|=c$ when $c$ is smaller than the diameter of the largest sphere that fits into $A$

I am especially interested in the case of rotational ellipsoids and there in the case where the major axis is the rotational axis.

By a “convex” manifold in this context I mean the boundary of a convex region of Euclidean space.