Definition:Normal Vector 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 normal vector $\map n s$ of $\alpha$ at $s$ is defined as:

$\map {\alpha} s = \map \kappa s \map n s$

where:

$\alpha$ denotes the second derivative of $\alpha$

That is:

$\map n s := \dfrac {\map {\alpha} s} {\norm {\map {\alpha} s} }$

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