Ive been trying to derive the equation of a hypocycloid. I am confused with one thing, in the hypocycloid is there a define direction of rotation and revolution of the smaller circle? (by direction I mean anticlockwise and clockwise). Because this seems to affect the answer derived.
Thank you for your help!!
Usual convention: \begin{align*} w &= a e^{i\theta} \\ c &= (a-b) e^{i\theta} \\ \frac{z-c}{w-c} &= e^{-i\phi} \\ a\theta &= b\phi \\ z &= (a-b)e^{i\theta}+b e^{-i\left( \frac{a-b}{b}\right) \theta} \end{align*}
$w$: point of contact
$c$: centre of blue circle
$z$: locus of the initial point of contact, i.e. the hypocycloid
$\theta$: angle swept by the centre $c$ or the contact point $w$
$\phi$: angle rotated by the blue circle relative to $w$