Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Vertices and Focal Curvatures of Space Curves   The {\em focal curve} of an immersed smooth curve $\gamma:s\mapsto
\gamma(s)$, in Euclidean space $\R^{m+1}$, consists of the centres of its
osculating hyperspheres. The focal curve may be parametrised in terms of the
Frenet frame of $\gamma$ (${\bf t},{\bf n}_1, ...,{\bf n}_m$), as
$C_\gamma(s)=(\gamma+c_1{\bf n}_1+c_2{\bf n}_2+...+c_m{\bf n}_m)(s)$, where the
coefficients $c_1,...,c_{m-1}$ are smooth functions that we call the {\em focal
curvatures} of $\gamma$. We discovered a remarkable formula relating the
Euclidean curvatures $\kappa_i$, $i=1,...,m$, 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 $k=1,...,m$, 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 $C_\gamma$.

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The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th