Osculating Sphere

The center of any Sphere which has a contact of (at least) first-order with a curve $C$ at a point $P$ lies in the normal plane to $C$ at $P$. The center of any Sphere which has a contact of (at least) second-order with $C$ at point $P$, where the Curvature $\kappa>0$, lies on the polar axis of $C$ corresponding to $P$. All these Spheres intersect the Osculating Plane of $C$ at $P$ along a circle of curvature at $P$. The osculating sphere has center

$${\bf a}={\bf x}+\rho\hat{\bf N}+{\dot\rho\over\tau}\hat{\bf B}$$

where $\hat{\bf N}$ is the unit Normal Vector, $\hat{\bf B}$ is the unit Binormal Vector, $\rho$ is the Radius of Curvature, and $\tau$ is the Torsion, and Radius

$$R=\sqrt{\rho^2+\left({\dot\rho\over\tau}\right)^2},$$

and has contact of (at least) third order with $C$.

See also Curvature, Osculating Plane, Radius of Curvature, Sphere, Torsion (Differential Geometry)

References

Kreyszig, E. Differential Geometry. New York: Dover, pp. 54-55, 1991.