The rolling motion of a convex symmetric body on a horizontal plane is a classical problem(studied independently by many people, starting from Chaplygin, Appel and Korteweg). In the symmetric case, Chaplygin was the first who showed how that the full equations of motion can be reduced to a linear integrable system of two ODEs. A modern exposition of Chaplygin's results can be found in the very recent book by Cushman, Śniatycki and Duistermaat.
The problem of rolling motion on an inclined plane is, in general, nonintegrable (this problem was studied, in particular, by V.V. Kozlov in 1990s).
As for the tracing trajectories of the point of contact, you might be interested in this article (also available on arXiv) and the references therein. The authors discuss the case of a disc (i.e. a convex body of revolution) rolling on a horizontal plane. From the Introduction:
It appears that the point of contact performs the composite bounded motion: it periodically traces some closed curve which rotates as a rigid body with some constant angular velocity about the fixed point...