Why is it true that $(\cos (a \theta +b), \sin (a \theta +b), (c \theta +d))$ for $\theta \in [\theta_1,\theta_2]$ can always be written as $(\cos \alpha \theta' , \sin \alpha \theta', \beta \theta')$ for a suitable choice of $\theta'\in [\theta'_1,\theta'_2]$?
Update: The parametrization describes a spiral of constant speed.
I don't think it's true. For the arguments of cos and sin, you must have that $\theta' = \frac{1}{\alpha}(a\theta + b)$, but then this means that $\theta = \frac{1}{a}(\alpha\theta' - b)$. Then the third coordinate must be $\frac{c}{a}(\alpha\theta' - b) + d = \frac{c\alpha}{a}\theta' + (d - \frac{bc}{d})$. Now you pick $\beta = \frac{c\alpha}{a}$, but it's not so clear why $d - \frac{bc}{a}$ must vanish; you don't get to choose $a,b,c,$ or $d$ so you don't have any control over whether or not they vanish.