I know that for a cycloid with radius $R$ and time $t$, it can be defined as $x = R(t - \sin t)$ and $y = R(1 - \cos t)$. However what if it's not at unit speed and we have a speed $z$/sec such that after $t$ seconds the centre is at $(tz, R)$.
2026-02-23 13:06:46.1771852006
How can I find the formula for a cycloid with a given speed?
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Consider the following parametrisation of the cycloid: $$\left\lbrace ~ \begin{aligned} x(\varphi) &= R ( \varphi - \sin\varphi ) \\ y(\varphi) &= R ( 1 - \cos \varphi ) \\ \end{aligned} \right .$$ What happens if you now choose $\varphi = z t$?