Let $\alpha: I \mapsto \mathbb{R^3}$ be a regular curve with non vanishing curvature. Check that if $t_0 \in I$ is fixed, it's possible to choose a cartesian coordinate system and reparameterize the curve by it's arclength $s$ such that at a neighborhood of $t_0$, $\alpha$ is approximately of the form:
$$\delta(s)=\left(s, \frac{k_0}{2}s^2,\frac{k_0 \tau_0}{6}s^3 \right)$$
where $k_0$ and $\tau_0$ are the curvature and torsion of $\alpha$ at $t_0$.
Now, I know what the local canonical form of a unit speed curve is, but I couldn't reparameterize it to get to the form above (and I don't know why reparameterizing it would lead to a result even simpler).
Edit: If $\alpha$ was unit speed, then using the usual approach to approximating an unit speed curve:
$$\alpha(s) \approx \left(s - \frac{k^2(0)}{6}s^3,\frac{k(0)}{2} s^2 + \frac{k'(0)}{6}s^3, -\frac{k(0)\tau(0)}{6}s^3\right)$$
Since it's not, let $\beta(s)$ be the reparameterization required, then:
$$\alpha(\beta(s)) \approx \alpha(\beta(0)) + (\alpha(\beta(0)))'s + (\alpha(\beta(0)))''\frac{s^2}{2} + (\alpha(\beta(0)))'''\frac{s^3}{6}$$
but that isn't very helpful. How can I proceed?