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.