A Doubt regarding Cycloid

Question

If we consider a cycloid made by a wheel. Then will the cycloid intersect the wheel when the wheel touches the topmost point of the cycloid? Thus will the radius of curvature be same to that of the wheel at that point?

Answer 1

From the parametric equations

$$\begin{align}x&=R(\theta-\sin\theta),\\y&=R(1-\cos\theta)\end{align}$$

the radius of curvature is

$$\frac{R((1-\cos\theta)^2+\sin^2\theta)^{3/2}}{|(1-\cos\theta)\cos\theta-\sin^2\theta|}=2^{3/2}R\sqrt{1-\cos\theta}.$$