How to change the parametric equations of the curtate cycloid to set the initial point

Question

I know that the parametric equations of the curtate cycloid of radius b and fixed point at the distance $a<b$ from the center of the circle are $$x(t)=at-b\cdot\sin{t}$$ $$y(t)=a-b\cdot\cos{t}$$ But with these equations the starting point is under the centre of the circle. If I want it above the centre, how can I modify these two equations? Do I only have to change the second in $$y(t)=a+b\cdot\cos{t}$$ or may I have to change something else?

Answer 1

If you start with the point above the center, as the wheel rolls it move forward faster at the start, so the equations should be $$x(t)=at+b\sin t\\y(t)=a+b\cos t$$ This corresponds to making the change $t \to t+\pi$ to rotate the circle, then resetting the zero of $x$ to where $a \pi$ was.