I'm currently studying differential geometry, and I'm trying to solve this proof-based exercise I've got assigned.
- Show that if α is a parametrised regular curve in R, then there exists a reparametrization β of α such that $| β'(t) |=1$.
Well, I know that all reparametrizations of regular curves are regular themlselves. So, β is regular.
I'm having more trouble with $| β'(t) |=1$. I know that if a derivative is 1, then the variation is constant through all values of t.
Does $| β'(t) |=1$ hold any special property? Can I relate it to β being unitary? Or is it as simple as saying that since α is regular, then there are infinite possibilities of reparametrization and one must be $| β'(t) |=1$ ?