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

Sources

• 2016: Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (2nd ed.): $1$-$5$: The Local Theory of Curves Parametrized by Arc Length