Cycloid rolls on another identical Cycloid A common cycloid with parametric equations $$ x = r ( t- \sin t - \pi) \, , y= (3+\cos t) r $$ with center point P $(0,4r)$ of its base at beginning of motion rolls on another identical but fixed cycloid without slipping: $$ x = r ( t- \sin t -\pi), \quad y= (1- \cos t) r $$ Find locus of P. Sorry about poor quality sketch. ![Rolling Cycloid](

At the point $P(t)$ on the second cycloid at parameter $t$, the slope is $\sin(t)/(1-\cos(t))$. The slope at the corresponding point on the first cycloid is $-\sin(t)/(1-\cos(t))$. We must translate $Q(t)$ to $P(t)$ and rotate by angle $\theta$ so that the slopes coincide: $\theta = 2 \arctan(\sin(t)/(1-\cos(t)))$. I get for the path of your point: $$ r [ \left( t-\pi \right) \cos \left( t \right) +t-\pi-2\,\sin \left( t \right) , \left( \pi-t \right) \sin \left( t \right) -2\,\cos \left( t \right) +2] $$

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