This paper introduces and analyzes several surface simplification methods for point-sampled geometry, including incremental and hierarchical clustering, iterative simplification, and particle simulation. These methods work directly on the point cloud without intermediate tessellation, and they employ local variation estimation and quadric error metrics to reduce approximation error and concentrate samples in regions of high curvature. The paper also presents a new method for computing numerical and visual error estimates for point-sampled surfaces. The algorithms are fast, easy to implement, and produce high-quality surface approximations, demonstrating the effectiveness of point-based surface simplification. The paper compares the quality, computational efficiency, and memory overhead of different techniques, providing guidance for users to choose the appropriate method for their specific applications. The methods are particularly useful for densely sampled organic shapes from 3D acquisition, iso-surface extraction, or sampling of implicit functions, but are not suitable for surfaces designed in specific surface representations like low-resolution polygonal CAD data.This paper introduces and analyzes several surface simplification methods for point-sampled geometry, including incremental and hierarchical clustering, iterative simplification, and particle simulation. These methods work directly on the point cloud without intermediate tessellation, and they employ local variation estimation and quadric error metrics to reduce approximation error and concentrate samples in regions of high curvature. The paper also presents a new method for computing numerical and visual error estimates for point-sampled surfaces. The algorithms are fast, easy to implement, and produce high-quality surface approximations, demonstrating the effectiveness of point-based surface simplification. The paper compares the quality, computational efficiency, and memory overhead of different techniques, providing guidance for users to choose the appropriate method for their specific applications. The methods are particularly useful for densely sampled organic shapes from 3D acquisition, iso-surface extraction, or sampling of implicit functions, but are not suitable for surfaces designed in specific surface representations like low-resolution polygonal CAD data.