Orientation preserving reparametrisation

Question

I'm currently taking a course in differential geometry, in which we are looking at curvature/torsion etc. In a couple of proofs provided in the lecture notes, the statement 'Without loss of generality, assume that $\tilde{f}:[\tilde{x},\tilde{y}] \to \mathbb{R}^n$ is an orientation preserving unit-speed reparametrisation of $f:[x,y]\to\mathbb{R}^n$.' is used.

My question is, can we make this assumption? That is, will it always be possible to find a reparametrisation that preserves orientation?

Note: We can assume that $f$ is sufficiently smooth etc.

Answer 1

Let $s(t) = \int_x^t |f'(u)| \, du$ be the arc-length function. Then $\tilde{f} = f \circ s^{-1}$ is the reparameterization. Note that by the chain rule: $$ \tilde{f} \, ' (t) = f'(s^{-1}(t)) \, \cdot (s^{-1})'(t) $$ By the Inverse Function Theorem, $$(s^{-1})'(t) = \frac{1}{s'(s^{-1}(t))} = \frac{1}{|f'(s^{-1}(t))|}$$ Therefore $\tilde{f} \, ' (t) = \frac{f'(s^{-1}(t))}{|f'(s^{-1}(t))|}$. This is the unit vector pointing in the same direction as $f'$, so it preserves orientation.