Smoothing and untangling triangular meshes on surfaces

In this work it has been developed the basis of shape quality metrics for triangular meshes on surfaces. It has also been worked out a simultaneous smoothing-untangling procedure based on the ideas introduced by Escobar. This procedure is designed through the minimization of an objective function (inverse of the quality of the mesh). We have increased the computational efficiency of the original method by adding a new algorithm for the search of the optimal projection plane. Moreover we have carried out a more robust discussion of the theoretical construction of the objective function for this new algorithm. Finally, several examples have been presented in order to illustrate the capabilities of the proposed method.