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This paper describes a new statistical approach for watermarking mesh representations of 3-D graphical objects. A robust digital watermarking method has to mitigate among the requirements of watermark invisibility, robustness, embedding capacity and key security. The proposed method employs a mesh propagation distance metric procedure called the fast marching method (FMM), which defines regions of equal geodesic distance width calculated with respect to a reference location on the mesh. Each of these regions is used for embedding a single bit. The embedding is performed by changing the normalized distribution of local geodesic distances from within each region. Two different embedding methods are used by changing the mean or the variance of geodesic distance distributions. Geodesic distances are slightly modified statistically by displacing the vertices in their existing triangle planes. The vertex displacements, performed according to the FMM, ensure a minimal surface distortion while embedding the watermark code. Robustness to a variety of attacks is shown according to experimental results.