Figure 1 / Decomposition of the motion of a rigid object. A) The positions at two sequential times (t1 and t2) of a line that is translating and rotating. B) Examples of the infinite number of physical movements that could underlie the sequential positions illustrated in (A). Although the center of rotation (black dot) can be anywhere, the angular speed of rotation (arrows) will always be constant for any possible decomposition of rigid movement giving rise to such an image sequence.

Figure 2 / Comparison of the perceived and predicted centers of rotation of a line that is translating and rotating. A) Example of a test stimulus (left) in which the stimulus line moves from the upper left downward to the right across a circular aperture while rotating around a fixed center on the line. Subjects were asked to report the perceived center of rotation at the moment the moving line became coincident with the dotted line (which immediately disappeared); they did so by marking the appropriate point with a dot in the response box on the right. B) Over time, the local maxima of the probability distributions of the possible sources of the stimulus in (A) define a cycloid (red circles). C) The sequential positions of the centers of rotation reported by subjects likewise fall along a cycloid with one cusp. The dots in panel show the raw data from the 5 subjects, and the diamonds in right panel the average position of the perceived centers of rotation at different times in the presentation; the solid red lines are the responses predicted by the theory (see panel B). The blue line indicates the actual trajectory of the center of rotation of the stimulus line, and the blue circles the times when subjects made their judgments.