We consider the piecewise linear approximation of saddle functions of the form (f(x,y)=ax^2-by^2 ) under the (L_ ) error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing the vertices of the approximation to move away from the graph of the function.
Bertram, Barnes, Hamann, Joy, Pottmann, and Wushour @cite_1 have extended this approach to an arbitrary bivariate function @math , by taking optimal local approximations on suitably defined patches and stitching'' them together at the patch boundaries. (The setting of this paper actually somewhat different: the bivariate function @math is given as a set of scattered data points.)
@cite_1
2014084922
We present an efficient algorithm to obtain a triangulated graph surface for scattered points (xiyi)T, i=1,…,n, with associated function values fi. The maximal distance between the triangulated graph surface and the function values is measured in z-direction (z=f(x,y)) and lies within a user-defined tolerance. The number of triangles is minimized locally by adapting their shapes to different second-degree least squares approximations of the underlying data. The method consists of three major steps: 1. subdividing the given discrete data set into clusters such that each cluster can be approximated by a quadratic polynomial within a prescribed tolerance; 2. optimally triangulating the graph surface of each quadratic polynomial; and 3. “stitching” the triangulations of all graph surfaces together. We also discuss modifications of the algorithm that are necessary to deal with discrete data points, without connectivity information, originating from a general two-manifold surface, i.e., a surface in three-dimensional space that is not necessarily a graph surface.