Interesting results for rolling and slipping

Rolling with slipping of a uniform rigid body of revolution on a rough horizontal or inclined plane in two-dimensional motion is a standard problem in elementary dynamics; in such motion each cross section of the body of revolution remains parallel to a fixed vertical plane that is taken to contain the line of greatest slope in the case of motion on an inclined plane. Consider similar two-dimensional motion inside a fixed cylinder with general, rather than circular, cross-sectional curve C. This much more difficult problem is considerably simplified if there is no gravity force. The translational and rotational velocities of the body are then found to depend only on the change in direction of the tangent to the curve C up to the current point of contact during the motion and not on the curve itself.