Any planar displacement of a rigid body can be uniquely characterized as a rotation about some fixed axis through a certain angle. These same rotational parameters can also describe the relative motion of two joint components and have been used to describe the rotational behavior of human body joints. This article derives estimates of these parameters and their standard errors, based on geometric arguments for the coordinates of markers observed during the rotation. This in turn identifies optimal placements of the experimental markers, which maximize the precision of the estimated center of rotation and the angle of rotation. This is the first statistical treatment of this problem. Previous approaches used deterministic error structures.