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 $\R^2$.

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