INTERACTIVE ROULETTE CURVES

Cycloid

$$ x_p(\theta) = (\theta + \sin(\theta))R $$ $$ y_p(\theta) = (1 + \sin(\theta))R $$

Epitrochoid

$${\displaystyle {\begin{aligned}&x(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \right)\\&y(\theta )=(R+r)\sin \theta -d\sin \left({R+r \over r}\theta \right)\end{aligned}}}$$

Hypotrochoid

$${\displaystyle {\begin{aligned}&x(\theta )=(R-r)\cos \theta +d\cos \left({R-r \over r}\,\theta \right)\\[1ex] &y(\theta )=(R-r)\sin \theta -d\sin \left({R-r \over r}\theta \right)\end{aligned}}}$$