This paper introduces a novel method to classify surfaces by their conformal structures. Surfaces in the same conformal class share the same uniformization metric, which induces constant Gaussian curvature everywhere on the surface. Under the uniformization metric, each homotopy class of a closed curves on the surface has a unique geodesic. The lengths of all closed geodesics form the geodesic spectrum. The map from the fundamental group to the geodesic spectrum completely determines the conformal structure of the surface.
We first compute the uniformization metric using discrete Ricci flow method, then compute the Fuchsian group generators, finally deduce the geodesic spectra from the generators in a closed form.
The method is rigorous and practical. Geodesic spectra is applied as the signature of surfaces for shape comparison and classification.