On Vertices, focal curvatures and differential geometry of space curves

The focal curve of an immersed smooth curve gamma : 0 -> gamma (theta), in Euclidean space Rm+1, consists of the centres of its osculating hyperspheres. This curve may be parametrised in terms of the Frenet frame of gamma (t, n(1), ..., n(m)), as C gamma(theta) = (gamma + c(1) n(1) + c(2)n(2) + (...) + c(m)n(m))(theta), where the coefficients c(1), ..., c(m-1) are smooth functions that we call the focal curvatures of gamma. We discovered a remarkable formula relating the Euclidean curvatures K-i, i = 1, ..., m, of gamma with its focal curvatures. We show that the focal curvatures satisfy a system of Frenet equations (not vectorial, but scalar!). We use the properties of the focal curvatures in order to give, for 1 = 1, ..., m, necessary and sufficient conditions for the radius of the osculating 1-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of gamma to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of gamma with the Frenet frame and the Euclidean curvatures of its focal curve C gamma.