Arbitrary refinement can be performed by inserting new mesh points at any desired location not limited to e.g. the mid-points of element edges. If the topology connecting the new mesh point is not limited to a fixed pattern, more general algorithms for the local remeshing will have to be employed. If the new point does not a priori belong to the mesh element which motivates the refinement, a point location method will be required to determine the element that contains the point. Coarsening meshes which are not of the hierarchical type by means of point removal has less practical relevance. The most important application is to collapse too close points and to coarsen a too fine surface mesh (see Section 3.5). Such fully flexible techniques are required to improve the geometrical quality of the mesh elements or to introduce anisotropy. An important refinement method for angle bounded triangulations by means of arbitrary point insertion is based on the Delaunay criterion and will be discussed in Section 5.6.