, Volume 66, Issue 3, pp 595-614

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Median Trajectories

  • Kevin BuchinAffiliated withTU Eindhoven
  • , Maike BuchinAffiliated withTU Eindhoven
  • , Marc van KreveldAffiliated withUtrecht University Email author 
  • , Maarten LöfflerAffiliated withUtrecht University
  • , Rodrigo I. SilveiraAffiliated withUniversitat Politècnica de Catalunya
  • , Carola WenkAffiliated withUniversity of Texas at San Antonio
  • , Lionov WiratmaAffiliated withParahyangan Catholic University


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.


Trajectories Geometric algorithms Homotopy