On Vertices, focal curvatures and differential geometry of space curves

The focal curve of an immersed smooth curve γ : θ ↦ γ (θ), in Euclidean space ℝ^m+1, consists of the centres of its osculating hyperspheres. This curve may be parametrised in terms of the Frenet frame of γ (t, n₁, . . . , n_m), as C_γ (θ) = (γ +c₁n₁+ c₂n₂ + • • • + c_mn_m)(θ), where the coefficients c₁, . . . , c_m-1 are smooth functions that we call the focal curvatures of γ . We discovered a remarkable formula relating the Euclidean curvatures κ_i , i = 1, . . . ,m, of γ 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 ℓ = 1, . . . ,m, necessary and sufficient conditions for the radius of the osculating ℓ-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of γ to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of γ with the Frenet frame and the Euclidean curvatures of its focal curve C_γ.