A procedure has been developed to improve polygonal surface mesh quality while maintaining the essential characteristics of the discrete surface. The surface characteristics are preserved by repositioning mesh vertices so that they remain on the original discrete surface. The repositioning is performed in a series of triangular-facet-based local parametric spaces. The movement of the mesh vertices is driven by a nonlinear numerical optimization process. Two optimization approaches are described, one which improves the quality of elements as much as possible and the other which improves element quality but also keeps the new mesh as close as possible to the original mesh.