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 
.
