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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 wedoi:10.4310/cis.2003.v3.n3.a2 fatcat:fkloh43245ervp7gsnv34oed7y