Osculating sphere

at a point $M$ of a curve $l$

The sphere having contact of order $n\ge 3$ with $l$ at $M$ (see Osculation). The osculating sphere can also be defined as the limit of a variable sphere passing through four points of $l$ as these points approach $M$. If the radius of curvature of $l$ at $M$ is equal to $\rho$ and $\sigma$ is the torsion, then the formula for calculating the radius of the osculating sphere has the form

$$R=\sqrt{{\rho }^2+\frac{1}{{\sigma }^2}{\left(\frac{d\rho }{ds}\right)}^2},$$

where $ds$ denotes the differential along an arc of $l$.

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