Periodicity has been recognized as an important cue for tasks like activity recognition and gait analysis. However, most existing techniques analyze periodic motions only in image coordinates, making them very dependent on the viewing angle. In this paper we propose a new technique for reconstructing periodic point trajectories in 3D given only their apparent trajectories in image coordinates from a single stationary camera. We show that this reconstruction is possible without performing a costly gradient descent-type optimization, and is based only on a single SVD. This new algorithm is shown to accurately reconstruct natural human motions, allowing them to be compared in 3D world coordinates, independent of the angle from which they were originally viewed.