The center of any Sphere which has a contact of (at least) first-order with a curve at a point lies
in the normal plane to at . The center of any Sphere which has a contact of (at least) second-order with
at point , where the Curvature , lies on the polar axis of corresponding to . All these
Spheres intersect the Osculating Plane of at along a circle of curvature at .
The osculating sphere has center
See also Curvature, Osculating Plane, Radius of Curvature, Sphere, Torsion (Differential Geometry)
References
Kreyszig, E. Differential Geometry. New York: Dover, pp. 54-55, 1991.