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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.

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    @RahulNarain: Thanks!2012-03-11

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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.