  In this paper we study the shape space of curves with values in a homogeneous
space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We
generalize the square root velocity framework to obtain a reparametrization
invariant metric on the space of curves in $M$. By identifying curves in $M$
with their horizontal lifts in $G$, geodesics then can be computed. We can also
mod out by reparametrizations and by rigid motions of $M$. In each of these
quotient spaces, we can compute Karcher means, geodesics, and perform principal
component analysis. We present numerical examples including the analysis of a
set of hurricane paths.
