Median Trajectories

We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and evaluate whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better.