Osculating Plane

The Plane spanned by the three points ${\bf x}(t)$, ${\bf x}(t+h_1)$, and ${\bf x}(t+h_2)$ on a curve as $h_1, h_2\to 0$. Let ${\bf z}$ be a point on the osculating plane, then

$$[({\bf z}-{\bf x}),{\bf x}',{\bf x}'']=0,$$

where $[{\bf A},{\bf B},{\bf C}]$ denotes the Scalar Triple Product. The osculating plane passes through the tangent. The intersection of the osculating plane with the Normal Plane is known as the Principal Normal Vector. The Vectors ${\bf T}$ and ${\bf N}$ (Tangent Vector and Normal Vector) span the osculating plane.

See also Normal Vector, Osculating Sphere, Scalar Triple Product, Tangent Vector