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
where is the unit normal vector, is the unit binormal vector, is the radius of curvature, and is the torsion, and radius
and has contact of (at least) third order with .