I am reading on Wikipedia that
''…Any regular curve may be parametrized by the arc length (the natural parametrization) and…''
I know that if $a(t) = (x(t),y(t),z(t))$ is a curve (say, smooth) then it is regular iff for all $t$: $a' (t) \neq 0$. I also know the definition of arc length:
The arc length of a curve $a$ between $t_0$ and $t$ is defined as
$$ l = \int_{t_0}^t |a'(t)|dt$$
But what is the parametrization of $a$ using its arc lenght?
Best Answer
Simply if $$\alpha: I\to \mathbb{R}^3$$ is a regular curve and the arc length is $$s(t)=\int_{t_0}^t |\alpha'|dt$$ Now we solve for $t$ as $t=t(s)$ to get the function $$t:J\to I$$So we define a new curve $\beta(s)=\alpha\circ t=\alpha(t(s))$ where $$\beta: J\to I \to \mathbb{R}^3 \\ |\beta'(s)|=|\frac{d\beta}{ds}|=|\frac{d\beta}{dt}. \frac{dt}{ds}|=|\frac{d\alpha(t(s))}{dt}. \frac{1}{\alpha'(t)}|=1$$