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We present a novel algorithm to segment a 3D surface mesh into visually meaningful regions. Our approach is based on an analysis of the local geometry of vertices. In particular, we begin with a novel characterization of vertices as convex, concave or hyperbolic based upon their discrete local geometry. Hyperbolic and concave vertices are considered potential feature region boundaries. We propose a new region growing technique starting from these boundary vertices leading to a segmentation of the surface that is subsequently simplified by a region-merging method. Experiments indicate that our algorithm segments a broad range of 3D models at a quality comparable to existing algorithms. Its use of methods that belong naturally to discretized surfaces and ease of implementation make it an appealing alternative in various applications.