Probabilistic properties of the elliptic motion

In this paper we consider the plane elliptic motion which occurs if the moving centrode is a circle of radius $r$ and the fixed centrode a circle of radius $2r$. Every point of the moving plane generates an ellipse in the fixed plane. Let a disk of radius $R$, $0 \le R < \infty$, concentric to the moving centrode be attached to the moving plane. If a point $P$ is chosen at random from this disk, then the area and the perimeter of the ellipse generated by $P$ are random variables. We determine the moments and the distributions of these random variables for the case that $P$ is uniformly distributed over the area of the disk.