Picture below shows two circles with centers M and N.
The smaller circle touches the bigger one inside and has half the diameter.
The smaller circle rolls inside the bigger one without friction.
Center N moves over the green circle.
The circles now touch in Q.
This implies an arc on the bigger circle of vp/180 degrees, if the radius is 1.
The smaller circle has half the radius, so it's rotation will be double the angle: 2v degrees.
Rotation is clockwise.
The new position P' of P after rotation is the sum of two rotations:
2. point P on the smaller circle, by 2v degrees clockwise
Now look at the picture below
The new position of P on the Y-scale becomes : Y = 0,5sin(v) + 0,5sin(-v) = 0,5sin(v) - 0,5sin(v) = 0
Conclusion: P moves over a straight line through center M.
On the web you may find this nice movie named the "crazy-circle-illusion".
You just read the explanation.