osculating plane

{{short description|Geometrical plane which second-order contacts a submanifold}}

{{More citations needed|date=January 2023}}

Image:frenet.svg, and the osculating plane (spanned by {{math|T}} and {{math|N}}).]]

In mathematics, 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 {{ety|la|osculari|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.{{Cite book |last=Do Carmo |first=Manfredo |title=Differential Geometry of Curves and Surfaces |isbn=978-0486806990 |edition=2nd |pages=18}}