I am studying some differential Geometry from Shifrin's differential Geometry notes.
From what I understand so far it seems like it is possible to unit-speed reparametrize a curve whenever the curve is regular.
In the notes that I am reading the parametrization of the circle is:
$\alpha = (a\cos{t},a\sin{t}), \implies \lVert \alpha '(t)\rVert = a$
And then it is mentioned that if we reparametrize the curve by
$\beta(s) = (a\cos(s/a),a\sin(s/a)) \implies \lVert\beta'(s)\rVert=1$
So my question how did the author derive that this exact reparametrization have unit speed?, He could impossibly just have guessed it..
And is there any General way to find a unit speed reparametrization of a parametrized curve?
Best Answer
The answer to your question is at the top of p. 8 of my notes. In the case of the circle as originally parametrized, the arclength, starting at $t=0$, is $s(t)=at$. So $t=s/a$. Thus, $\beta(s) = \alpha(s/a) = \big(a\cos(s/a),a\sin(s/a)\big)$ is a reparametrization by arclength. You can immediately check that $\|\beta'(s)\|=1$, but the general argument is in the notes there.