### Geometric Compression Using Riemann Surface Structure

Xianfeng Gu, Yalin Wang, and Shing-Tung Yau

Communications in Information and Systems, 3(3), 2004, pp. 171-182

This paper introduces a theoretic result that shows any surface in 3 dimensional Euclidean space can
be determined by its conformal factor and mean curvature uniquely up to rigid motions. This theorem disproves the
common belief that surfaces have three functional freedoms and immediately shows that one third of geometric data
can be saved without loss of information.
The paper develops a practical algorithm to losslessly compress geometric surfaces based on Riemann surface
structures. First we compute a global conformal parameterization of the surface. The surface can be segmented by
holomorphic flows, where each segment can be conformally mapped to a rectangle on the parameter plane, which is
guaranteed by circle-valued Morse theory. We construct a conformal geometry image for each segment, and record
conformal factor and dihedral angle for each edge. In this way, we represent the surface using only two functions
with canonical connectivity. We present the proofs of the theorems and the compression examples.