The MLS surface, used for modeling and rendering with point clouds, was originally defined algorithmically as the output of a particular meshless construction. We give a new explicit definition in terms of the critical points of an energy function on lines determined by a vector field. This definition reveals connections to research in computer vision and computational topology. Variants of the MLS surface can be created by varying the vector field and the energy function. As an example of such a generalization, we define a point-set surface for surfels (points equipped with normals). We also make an important technical observation: the procedures described in the literature to take points in space onto the MLS surface do not, in fact, produce points of the MLS surface. We describe a simple iterative procedure which does, for any of the variants.