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In many cases the surfaces of geometric models consist of a large number of triangles. Several algorithms were developed to reduce the number of triangles required to approximate such objects. Algorithms that measure the deviation between the approximated object and the original object are only available for special cases. We use the Hausdorff distance between the original and the simplified mesh as a geometrically meaningful error value which can be applied to arbitrary triangle meshes. We present a new algorithm to reduce the number of triangles of a mesh without exceeding a user defined Hausdorff distance between the original and simplified mesh. As this distance is parameterization independent, its use as error measure is superior to the use of the L ∞-Norm between parameterized surfaces. Furthermore the Hausdorff distance is always less than the distance induced by the L ∞-Norm. This results in higher reduction rates. Excellent results were achieved by the new decimation algorithm for triangle meshes that has been used in different application areas such as volume rendering, terrain modeling and the approximations of parameterized surfaces. The key advantages of the new algorithm are: it guarantees a user defined position dependent approximation error; it allows one to generate a hierarchical geometric representation in a canonical way; it automatically preserves sharp edges.