This next app allows the student to user sliders, buttons and checkboxes to adjust the parameters of an epicycloid, hypocycloid, epitrochoid or hypotrochoid, the equations for which are given by:
$x\left(\mathrm{\θ}\right)\=\left(Rplus;r\right)\cdot \mathrm{cos}\left(\mathrm{theta;}\right)plus;s\cdot L\cdot r\cdot \mathrm{cos}\left(\frac{Rplus;r}{r}\cdot \mathrm{theta;}\right)$
$y\left(\mathrm{\theta}\right)\=\left(Rplus;r\right)\cdot \mathrm{sin}\left(\mathrm{\theta}\right)L\cdot r\cdot \mathrm{sin}\left(\frac{Rplus;r}{r}\cdot \mathrm{\theta}\right)$
The curve is called an epicycloid when $s\=1$ and $L\=1$; a hypocycloid when $s\=1$ and $L\=1$; an epitrochoid when $s\=1$ and $L\ne 1$; and a hyptrochoid when $s\=1$ and $L\ne 1$.
The figure can also be animated by clicking on the "Play" button.



Fixed circle radius (R) =

Rolling circle radius (r) =

Start
End

Ratio ($L$) of Pen length/radius





