  We propose an optimization algorithm for computing geodesics on the universal
Teichm\"uller space T(1) in the Weil-Petersson ($W P$) metric. Another
realization for T(1) is the space of planar shapes, modulo translation and
scale, and thus our algorithm addresses a fundamental problem in computer
vision: compute the distance between two given shapes. The identification of
smooth shapes with elements on T(1) allows us to represent a shape as a
diffeomorphism on $S^1$. Then given two diffeomorphisms on $S^1$ (i.e., two
shapes we want connect with a flow), we formulate a discretized $W P$ energy
and the resulting problem is a boundary-value minimization problem. We
numerically solve this problem, providing several examples of geodesic flow on
the space of shapes, and verifying mathematical properties of T(1). Our
algorithm is more general than the application here in the sense that it can be
used to compute geodesics on any other Riemannian manifold.
