Definition:Osculating Plane of Curve Parameterized by Arc Length/3-Dimensional Real Vector Space

Definition
Let $\alpha : I \to \R^3$ be a smooth curve parameterized by arc length.

Let $s \in I$ be such that the curvature $\map \kappa s \ne 0$.

The osculating plane of $\alpha$ at $s$ is the linear span of:
 * $\set {\map t s, \map n s}$

where:
 * $\map t s$ is the unit tangent vector
 * $\map n s$ is the normal vector