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