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We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The computation of points on the surface is local, which results in an out-of-core technique that can handle any point set. We show that the approximation error is bounded and present tools to increase or decrease the density of the points, thus allowing an adjustment of the spacing among the points to control the error. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates.