Trochoid (point at distance \(d\) from the rolling circle center)

$$ x(\theta,\varphi)=R\,\theta - d\,\sin\!\bigl(\theta-\varphi\bigr),\qquad y(\theta,\varphi)=R - d\,\cos\!\bigl(\theta-\varphi\bigr). $$

Cycloid (special case \(d=R\), \(\varphi=0\))

$$ x(\theta)=R\!\left(\theta - \sin(\theta)\right),\qquad y(\theta)=R\!\left(1 - \cos(\theta)\right). $$

Epitrochoid (point at distance \( d \) on a circle of radius \( r \) rolling outside a fixed circle of radius \( R \))

$$ x(\theta) = (R+r)\cos(\theta) - d\cos\!\left(\frac{R+r}{r}\,\theta\right),\qquad y(\theta) = (R+r)\sin(\theta) - d\sin\!\left(\frac{R+r}{r}\,\theta\right). $$

Special case (epicycloid, \( d=r \))

$$ x(\theta) = (R+r)\cos(\theta) - r\cos\!\left(\frac{R+r}{r}\,\theta\right),\qquad y(\theta) = (R+r)\sin(\theta) - r\sin\!\left(\frac{R+r}{r}\,\theta\right). $$

Hypotrochoid (point at distance \( d \) on a circle of radius \( r \) rolling inside a fixed circle of radius \( R \))

$$ \begin{aligned} x(\theta) &= (R-r)\cos(\theta) \;+\; d\cos\!\left(\frac{R-r}{r}\,\theta\right),\\[4pt] y(\theta) &= (R-r)\sin(\theta) \;-\; d\sin\!\left(\frac{R-r}{r}\,\theta\right). \end{aligned} $$

Special case (hypocycloid, \( d = r \))

$$ \begin{aligned} x(\theta) &= (R-r)\cos(\theta) \;+\; r\cos\!\left(\frac{R-r}{r}\,\theta\right),\\[4pt] y(\theta) &= (R-r)\sin(\theta) \;-\; r\sin\!\left(\frac{R-r}{r}\,\theta\right). \end{aligned} $$