Vertex based data dependent triangulations (Q1183517)

The problem is studied to choose a triangulation of the convex hull of a set of points and data values at the points so that the resulting piecewise linear interpolating surface minimizes the average absolute error and is pleasing to the eye. Schemes called angles between normals (ABN) with \(\ell_ 1\) and \(\ell_ 2\) norms introduced by \textit{N. D. Dyn}, \textit{D. Levin}, and \textit{S. Rippa} [IMA J. Num. Anal. 10, No. 1, 137-154 (1990; Zbl 0699.65004)] and methods based on vertex local optimization ( piecewise linear analog of curvature, PLC) are discussed and compared. Initial triangulations are altered to that the costs (the costs were defined) are reduced. Experiments with a set of 100 data points are carried out, the resulting ABN, PLC and Delaunay surfaces and triangulations are shown, and absolute errors and numbers of iterations are tabulated. PLC offers better results for surfaces with a preferred direction.