The focal locus ∑xof an affine varietyXis roughly speaking the (projective) closure of the set of pointsOfor which there is a smooth point x ∈Xand a circle with centreOpassing through x which osculatesXinx. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map ∈ between the Euclidean normal bundle Nxand the projective ambient space (∈ sends the normal vectorO- x to its endpointO), and in this paper we address two general problems:.