On Vertices and Focal Curvatures of Space Curves

The {\em focal curve} of an immersed smooth curve $\gamma:s\mapsto \gamma(s)$,inEuclideanspace$\R^{m+1}$, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of $\gamma$ ($\mathbf{t},{\mathbf{n}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathbf{n}}_m$), as $C_{\gamma }(s)=(\gamma +c_1{\mathbf{n}}_1+c_2{\mathbf{n}}_2+\mathrm{.}\mathrm{.}\mathrm{.}+c_m{\mathbf{n}}_m)(s)$, where the coefficients $c_1,\mathrm{.}\mathrm{.}\mathrm{.},c_{m-1}$ are smooth functions that we call the {\em focal curvatures} of $\gamma$. We discovered a remarkable formula relating the Euclidean curvatures ${\kappa }_i$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},m$, 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 $k=1,\mathrm{.}\mathrm{.}\mathrm{.},m$, necessary and sufficient conditions for the radius of the osculating $k$-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 $C_{\gamma }$.