Optimal Triangulation of Saddle Surfaces

TL;DR

This paper investigates the optimal piecewise linear approximation of saddle surfaces under the L-infinity norm, showing that non-interpolating approximations can achieve smaller errors by adjusting vertex positions.

Contribution

It demonstrates that interpolating approximations are suboptimal and introduces a method for improving approximation accuracy by moving vertices away from the surface.

Findings

Interpolating approximations are not optimal for saddle surfaces.

Allowing vertices to move away from the surface reduces approximation error.

Optimal approximation involves vertex adjustments for minimal error.

Abstract

We consider the piecewise linear approximation of saddle functions of the form $f(x,y)=ax^2-by^2$ under the L-infinity 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.