Parametric equations of cycloid on a Ramp

The parametric equation of a cycloid generated by rolling a wheel of radius $a$ at a constant rate by an angle $\theta$ is

$$x = a (\theta-\sin{\theta}) \quad y=a (1-\cos{\theta})$$

Rotating the $(x,y)$ coordinate system to a new coordinate system $(x',y')$ by an angle $\phi$ is accomplished by the tranformation

$$x'=x \cos{\phi}+y \sin{\phi} \quad y'=-x \sin{\phi}+y \cos{\phi}$$

so that the equation of the cycloid here is, after some simplification:

$$x'=a (\sin{\phi}+\theta \cos{\phi}) - a \sin{(\theta+\phi)} \quad y' = a(\cos{\phi}-\theta \sin{\phi}) - a \cos{(\theta+\phi)}$$

Note this assume that the point $P$ begins on the ramp. Note also that $\phi = \pi/2-\alpha$, the angle of incline.

Here is a plot for a constant rate at the specified angle of incline, $\alpha=\pi/3$.