This work introduces a system of algorithms to compute period matrices for general
surfaces with arbitrary topologies. The algorithms are intrinsic to the geometry, and independent of
surface representations. The computation is efficient, stable and practical for real applications. The
algorithms are experimented on real surfaces including human faces and sculptures, and applied to
surface identification problems. It is the first work that is both theoretically solid, and practically
robust and accurate to handle real surfaces with arbitrary topologies.