Definition:Curvature 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 $s \in I$.

The curvature of $\alpha$ at $s$ is defined as:
 * $\map \kappa s := \norm {\map {\alpha' '} t}$

where:
 * $\alpha ' '$ denotes the second derivative of $\alpha$
 * $\norm \cdot$ denotes the Euclidean norm on $\R^3$

Also known as
Some sources use the spelling parametrized.

Also see

 * Definition:Curvature: for curves in $\R^2$.