Definition:Arc Length of Regular Curve/3-Dimensional Real Vector Space

Definition

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

Let $t_0 \in I$.

Then the arc length of $\alpha$ from $t_0$ is defined as:

$\ds \map s t := \int_{t_0}^t \norm {\map {\alpha '} t} \rd t$

where:

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

$\norm \cdot$ denotes the Euclidean norm on $\R^3$

Sources

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