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