, particularly in differential geometry
, an osculating plane
is a plane
in a Euclidean space
or affine space
which meets a submanifold
at a point in such a way as to have a second order of contact
at the point. The word osculate
is from the Latin osculatus
which is a past participle
, meaning to kiss
. An osculating plane is thus a plane which "kisses" a submanifold.
The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors. See also Differential geometry of curves#Special Frenet vectors and generalized curvatures.