  Given two simplicial complexes in R^d, and start and end vertices in each
complex, we show how to compute curves (in each complex) between these
vertices, such that the Fr\'echet distance between these curves is minimized.
As a polygonal curve is a complex, this generalizes the regular notion of weak
Fr\'echet distance between curves. We also generalize the algorithm to handle
an input of k simplicial complexes.
  Using this new algorithm we can solve a slew of new problems, from computing
a mean curve for a given collection of curves, to various motion planning
problems. Additionally, we show that for the mean curve problem, when the k
input curves are c-packed, one can (1+epsilon)-approximate the mean curve in
near linear time, for fixed k and epsilon.
  Additionally, we present an algorithm for computing the strong Fr\'echet
distance between two curves, which is simpler than previous algorithms, and
avoids using parametric search.
