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Triangle meshes are widely used in representing surfaces in computer vision and computer graphics. Although 2D image processing-based edge detection techniques have been popular in many application areas, they are not well developed for surfaces represented by triangle meshes. This paper proposes a robust edge detection algorithm for triangle meshes and its applications to surface segmentation and adaptive surface smoothing. The proposed edge detection technique is based on eigen analysis of the surface normal vector field in a geodesic window. To compute the edge strength of a certain vertex, the neighboring vertices in a specified geodesic distance are involved. Edge information are used further to segment the surfaces with the watershed algorithm and to achieve edge-preserved, adaptive surface smoothing. The proposed algorithm is novel in robustly detecting edges on triangle meshes against noise. The 3D watershed algorithm is an extension from previous work. Experimental results on surfaces reconstructed from multi-view real range images are presented.