  In this paper, we present a new method for computing approximate geodesic
distances. We introduce the wave method for approximating geodesic distances
from a point on a manifold mesh. Our method involves the solution of two linear
systems of equations. One system of equations is solved repeatedly to propagate
the wave on the entire mesh, and one system is solved once after wave
propagation is complete in order to compute the approximate geodesic distances
up to an additive constant. However, these systems need to be pre-factored only
once, and can be solved efficiently at each iteration. All of our tests
required approximately between 300 and 400 iterations, which were completed in
a few seconds. Therefore, this method can approximate geodesic distances
quickly, and the approximation is highly accurate.
