The Domain of a Point Set Surface

It is useful to be able to define a two-dimensional point-set surface determined by a point cloud. One popular definition is Levin's MLS surface. This surface is defined on a domain which is a three-dimensional subset of R3, a narrow region around the input point cloud. If we were to extend the definition outside the domain, we would produce components of the surface which are far from the point cloud. This is important in practice, since when moving points onto the MLS surface, we need to begin with an initial guess which is within the domain. We visualize the domain in two dimensions, and explain why it is so narrow. We also consider two MLS variants which can be defined on a wider domain without producing spurious surface components. One is efficient and works well except near sharp corners. The other is computationally expensive but seems to work well everywhere.