These two circles have radius \(1\). Imagine the green circle rolling round the blue circle, starting with the green cross on the blue cross. How do the lengths of the two highlighted arcs compare?

Since the green circle is rolling without slipping, the arcs must be the same length

When the pink angle is \(\theta\), what are the coordinates of the centre of the green circle? What is the red angle?

What is the blue angle?

What are the lengths of the yellow and pink line segments?

What are the coordinates of the green point?

$$\begin{align*} \sin(2\theta-90^\circ)&=-\cos(2\theta)\\[4pt] \cos(2\theta-90^\circ)&=\sin(2\theta)\\[4pt] \Rightarrow \text{green cross is at }&(2\cos\theta-\cos2\theta,\,2\sin\theta-\sin2\theta) \end{align*}$$

What parametric form of the equation of the locus of the green point does this give?

$$\begin{align*} x&=2\cos\theta-\cos2\theta\\[4pt] y&=2\sin\theta-\sin2\theta \end{align*}$$