Curve arc length parametrization definition

Question

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

http://homepage.smc.edu/kennedy_john/ArcLengthParametrization.pdf

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?

Answer 1

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 $\Bbb R$.

Answer 2

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.