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.

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Partition of an interval definition
Arc length definition

Tag	`0X7G61297X`
URI	`m2://0X7G61297X`
Kind	definition
Writing	W1 stated
MSC	26A42, 53A04
Defines	Arc length
Uses	Partition of an interval

Arc length

Definition

Remarks

Reading path

Further reading

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