Inside a circle or radius {jatex options:inline}2r{/jatex} is a circle radius {jatex options:inline}r{/jatex} as shown. P is a fixed point on the circumference of the smaller circle.

The smaller circle starts to roll anticlockwise around the interior of the larger circle. While the small circle rolls anticlockwise, it rotates clockwise about its own centre. In fact if the small circle rolls once about the the large circle, it rolls a length {jatex options:inline}L=2 \pi (2r)=4 \pi r{/jatex} so will turn about an angle {jatex options:inline}\frac{4 \pi r}{r}= 4 \pi{/jatex} about its centre. In fact the small circle turns about its own centre through twice the angle that it turns about the centre of the large circle.

The angle OCP is isosceles and taking the centre of the large circle as the origin, the coordinates of P are {jatex options:inline}(rcos \alpha + r cos \alpha , rsin \alpha - r sin \alpha )=(2r cos \alpha , 0){/jatex}.

Hence the {jatex options:inline}y{/jatex} coordinate of P is constant and the point P remains on the {jatex options:inline}x{/jatex} axis.