Reparameterize the helix, $\alpha(t) = (a*\cos(t),a*\sin(t),b*t)$ by arc length

Question

Reparameterize $\alpha(t) = (a*\cos(t),a*\sin(t),b*t)$ by arc length. That is, give an equivalent parameterization $\gamma$ of the helix such that $|\gamma'(t)|=1$

If $\gamma(t)= (a*\cos(t/\sqrt2a),a*\sin(t/\sqrt2a),t/\sqrt2)$

$\rightarrow$ $\gamma'(t)= (\sqrt2*\cos(t/\sqrt2a),\sqrt2*\sin(t/\sqrt2a),1/\sqrt2)$

$|\gamma'(t)|= (\sqrt2*\cos(t/\sqrt2))^2 + (\sqrt2*\sin(t/\sqrt2))^2 + (1/\sqrt2)^2$

= $1/2 + 1/2 = 1$

Is this correct? And to get from one parameterization to the other I would use a homeomorphism that sends $t \rightarrow t/\sqrt2$. Correct?

Answer 1

No, the re-parameterization is $t\to \frac{t}{\sqrt{2}a}$ if $b=a$; otherwise the re-parameterization is $t\to \frac{t}{|(a,b)|}$.