Optimal triangulations for piecewise linear approximations of non-convex variable products

Abstract

We show optimal triangulations for piecewise linear (PWL) approximations of indefinite quadratic functions over the plane. Optimal triangulations have minimum triangle density while allowing a PWL approximation that fulfills a prescribed error bound measured in the L-infinity norm. In 2000, Pottmann et al. proved optimal triangulations for PWL interpolations and conjectured that these are also optimal for general PWL approximations. This conjecture was refuted in 2018 by Atariah et al., who allowed a constant deviation at the vertices of the triangles and decreased the triangle density by roughly 3%, though they left open whether their construction was optimal. In this paper, we resolve this open question: allowing varying deviations at vertices reduces the optimal triangle density by 25% compared to Atariah et al., and we prove this is globally optimal. We thus show that the potential of general PWL approximations is significantly lower for indefinite than for definite quadratic functions, where the triangle density can be halved when allowing general approximations compared to interpolations. Furthermore, we prove that among parallelogram tilings -- triangulations built from translated copies of a triangle and its point-reflection -- the constant-deviation construction of Atariah et al. is optimal when continuity of the PWL approximation is required. We conjecture that this optimality extends to all continuous triangulations, not just those based on parallelograms.