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?
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Perhaps you would like to tell us what these equations are ... & then repeat the title so we know what your question is ? – Donald Splutterwit Jun 19 '17 at 20:24
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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.
Ross Millikan
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Ok, thanks.. another question, if possible: how does it change the formula to calculate its length if the starting point change as described? It is not equal, isn't it? – Jun 20 '17 at 00:35
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The new $t$, which is $\phi$ in the Mathworld article is $t-\pi$, so the formula given there would give you the arc length from $t=\pi$ to somewhere. That is because $t=\pi$ is the bottom of an arc. If you want the arc length from $t_1$ to $t_2$ you can use that formula to get the length from $t=\pi$ to $t=t_1$ and from $t=\pi$ to $t=t_2$ and subtract. – Ross Millikan Jun 20 '17 at 01:16