Can a rotating circle fill all sides of a rotated rectangle?

Question

Consider we have a fixed rotating circle and a rotating rectangle which is forced to be tangent with the circle. Does circle travel all points of rectangle's Perimeter?

Answer 1

If the circle is on the outside of the rectangle and they are constantly touching each other without slipping then yes. Due to the fact that the circle is always tangent to the rectangle and that the circle and rectangle are both rotating, a $\theta$ degree rotation for the circle corresponds to a $\theta*q$ degree rotation for the rectangle where $q = \frac{P_{Circle}}{P_{Rectangle}}$.

Because the rectangle and circle are constantly tangent to each other without slipping, the circle will eventually touch all points on the rectangle's perimeter. And as martycohen mentioned this is true if the rectangle were to be replaced by any convex polygon.

But, if the circle is on the inside of the rectangle the answer is no because the corners of the rectangle(or the polygon in question) would never be touched.