Arc length

Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be continuous. The arc length of $\gamma$ is $$L(\gamma) = \sup_{P} \sum_{i=1}^{k} \| \gamma(t_i) - \gamma(t_{i-1}) \|,$$ where the supremum runs over all partitions $P = (t_0, \dots, t_k)$ of $[a, b]$. The curve $\gamma$ is rectifiable if $L(\gamma) < \infty$.

Remarks

Each sum is the length of a polygonal path through points of the curve. Adding points does not decrease this length. If $\gamma$ is continuously differentiable, then $L(\gamma) = \int_a^b \| \gamma'(t) \| \, dt$. For the graph of a $C^1$ function $f$ on $[a, b]$, this gives $L = \int_a^b \sqrt{1 + f'(x)^2} \, dx$. Some continuous curves are not rectifiable, for example the graph of the Weierstrass function.

Explanations

L1 human

The arc length of a curve is the length of a string laid along it. We approximate the curve by straight segments and take the best approximation.