Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I did some assignments related to curve arc length parametrization.

But what I can't seem to find online is a formal definition of it.

I've found procedures and ways to find a curve's equation by arc length parametrization, but I'm still missing a formal definition which I have to write in my assignment.

I saw many links related to the topic

but they all seem too long and don't provide a short, concise definition.

Could anyone help me writing a formal definition of curve arc length parametrization?

share|cite|improve this question
$\gamma:[a,b]\to\Bbb R^n,\|\gamma\,'\|=1$? – anon May 22 '12 at 16:59
Want to write it up as an answer? So I can accept it :). – Tool May 22 '12 at 17:02
up vote 3 down vote accepted

A curve (of finite length) with parametrization $\gamma:[a,b]\to\Bbb R^n$ is said to be the arclength, or natural, parametrization if the speed $\|\gamma'\|=1$ is always unity. Same thing for infinite length, you just need a larger interval as domain, like $[0,\infty)$ or all of $\Bbb R$.

share|cite|improve this answer

Suppose $\gamma:[a,b]\rightarrow {\Bbb R}$ is a smooth curve with $\gamma'(t) \not = 0$ for $t\in[a,b]$.

Define $$s(t) = \int_a^t ||\gamma'(\xi)||\,d\xi$$ for $t\in[a,b]$. This function $s$ has a positive derivative, so it possesses a differentiable inverse. You can use it to get a unit-speed reparametrization of your curve.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.