Definition:Curve Parameterized by Arc Length/3-Dimensional Real Vector Space

Definition

Let $\alpha : I \to \R^3$ be a smooth curve.

$\alpha$ is said to be parameterized by arc length if and only if:

$\forall t \in I : \norm {\map {\alpha'} t} = 1$

where:

$\alpha '$ denotes the derivative of $\alpha$

$\norm {\, \cdot \,}$ denotes the Euclidean norm on $\R^3$

Also known as

Some sources use the spelling parametrized.

Sources

• 2016: Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (2nd ed.): $1$-$3$: Regular Curves; Arc Length