Osculating sphere

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

The sphere having contact of order $n\geq3$ 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 $l$.

References

[a1]	R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1979) pp. 39
[a2]	D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988) pp. 25