A novel technique for multiscale curvature computation on a smoothed 3-D surface is presented. This is achieved by iteratively convolving local parameterizations of the surface with 2-D Gaussian filters. In our technique, semigeodesic coordinates are constructed at each vertex of the mesh which becomes the local origin. A geodesic from the origin is first constructed in an arbitrary direction such as the direction of one of the incident edges. The smoothing eliminates surface noise and small surface detail gradually and results in gradual simplification of the object shape. The surface Gaussian and mean curvature values are estimated accurately at multiple scales together with curvature zero-crossing contours. The curvature values are then mapped to colors and displayed directly on the surface. Furthermore, maxima of Gaussian and mean curvatures are also located and displayed on the surface. These features have been utilized by later processes for robust surface matching and object recognition. Our technique is independent of the underlying triangulation and is also more efficient than volumetric diffusion techniques since 2-D rather than 3-D convolutions are employed. Another advantage is that it is applicable to incomplete surfaces which arise during occlusion or to surfaces with holes.