Long and Thin Triangles Can Be Good for Linear Interpolation

Given a set of data points in R² and corresponding data values, it is clear that the quality of a Piecewise Linear Interpolation Surface (PUS) over triangles depends on the specific triangulation of the data points. In this paper, the question of what are good triangles (and triangulations) for linear interpolation is studied further. First, the model problem of constructing optimal triangulations for interpolating quadratic functions by PLIS is considered. Next, a new interpretation of existing error bounds for interpolating general smooth functions by PUS is studied. The conclusion is that triangles should be long in directions where the magnitude of the second directional derivative of F is small and thin in directions where the magnitude of the second directional derivative of F is large.