Least-squares meshes

In this paper we introduce least-squares meshes: meshes with a prescribed connectivity that approximate a set of control points in a least-squares sense. The given mesh consists of a planar graph with arbitrary connectivity and a sparse set of control points with geometry. The geometry of the mesh is reconstructed by solving a sparse linear system. The linear system not only defines a surface that approximates the given control points, but it also distributes the vertices over the surface in a fair way. That is, each vertex lies as close as possible to the center of gravity of its immediate neighbors. The least-squares meshes (LS-meshes) are a visually smooth and fair approximation of the given control points. We show that the connectivity of the mesh contains geometric information that affects the shape of the reconstructed surface. Finally, we discuss the applicability of LS-meshes to approximation of given surfaces, smooth completion and mesh editing.