Local optimization of triangular surface meshes for general quadrics in lp norm

For a set S of distinct points in the x-y plane and a 3-dimensional surface with the general quadratic equation f ey dx cxy by ax z + + + + + = 2 2 we study the problem of finding a locally optimal triangulation of S for the linear approximation of the surface under the Lp norm. We show that the Lp norm error is independent of the translation and rotation by 180 degrees of a single triangle in the x-y plane. This observation allows us to substantially simply the local optimization procedure, and in particular allows us to explicitly compute the separation curves. Locally optimal triangulations are important in the data dependent approximation methods. The paper generalizes earlier results by P. Desnogues and O. Devillers (presented at the Canadian Conference on Computational Geometry CCCG ‘95), who studied optimal local triangulations for the unit hyperbolic paraboloid in the L2 norm error.