The use of polygonal meshes, especially triangle meshes, is manifold, but a lot of algorithms require the mesh to be structured in a certain way and cannot be applied to an arbitrarily structured mesh. The process of replacing an arbitrarily structured mesh by a structured one is called remeshing and the most important class of structured triangle mesh are those with subdivision connectivity. Given an initial triangle mesh and a coarse base mesh with the same topology which serves as a parameter domain, it is possible to construct a one-to-one mapping between both meshes. This parameterization can then be used to set the vertices of a structured triangle mesh with subdivision connectivity such that this remesh approximates the initial mesh. The efficient multiresolution algorithms that can be applied to this remesh include wavelet decomposition, level-of-detail rendering, editing and progressive transmission.

Over the last years the problem of fairing triangle meshes has received a lot of attention. The need for these methods ranges from technical applications, where the noise that is due to measurement errors has to be removed from measured data, to entertainment applications, that require triangulated 3D models with a pleasing visual appearance. The usual approach in mesh fairing is to move the vertices of the mesh such that a certain energy functional is minimized. However, these methods cannot be applied whenever the position of the original data points must not be changed, e.g. in numerical simulations or surface interpolation. The only parameter that is left to changes is the triangulation of the data points itself. By sequentially swapping edges such that a global fairing functional is reduced by each swap, a given triangulation can be optimized without changing the position of its vertices but only the connectivity among them.