An algebraic method for smoothing surface triangulations on a local parametric space

This paper presents a new procedure to improve the quality of triangular meshes defined on surfaces. The improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes a certain objective function. This objective function is derived from algebraic quality measures of the local mesh (the set of triangles connected to the adjustable or free node). If we allow the free node to move on the surface without imposing any restriction, only guided by the improvement of the quality, the optimization procedure can construct a high-quality local mesh, but with this node in an unacceptable position. To avoid this problem the optimization is done in the parametric mesh, where the presence of barriers in the objective function maintains the free node inside the feasible region. In this way, the original problem on the surface is transformed into a two-dimensional one on the parametric space. In our case, the parametric space is a plane, chosen in terms of the local mesh, in such a way that this mesh can be optimally projected performing a valid mesh, that is, without inverted elements. Several examples and applications presented in this work show how this technique is capable of improving the quality of triangular surface m