Constructive quasi-uniform sequences over triangles

TL;DR

This work addresses constructing high-quality point samples on arbitrary triangles by introducing the Voronoi-guided greedy packing (VG) algorithm, which selects the next point from a finite Voronoi-based candidate set to maximize separation while maintaining coverage. The principal theoretical contribution proves that the mesh ratio satisfies $\rho(P_n;T) \le 2$ after finitely many steps, achieving the optimal quasi-uniform bound, with an explicit bound on the number of steps $K$ when starting from the triangle vertices. The authors further analyze triangular low-discrepancy sequences, proving their quasi-uniformity (with bounds dependent on triangle geometry) and compare these sequences to VG through extensive numerical experiments, including RBF interpolation on triangles. The results show VG provides a practical, scalable, and robust method that rivals the best quasi-uniform grids while outperforming several low-discrepancy constructions in mesh-variance and interpolation stability, thereby offering a reliable tool for mesh generation, scattered data approximation, and kernel-based interpolation on triangular domains.

Abstract

In this paper, we develop constructive algorithms for generating quasi-uniform point sets and sequences over arbitrary two-dimensional triangular domains. Our proposed method, called the \emph{Voronoi-guided greedy packing} algorithm, iteratively selects the point farthest from the current set among a finite candidate set determined by the Voronoi diagram of the triangle. Our main theoretical result shows that, after a finite number of iterations, the mesh ratio of the generated point set is at most~2, which is known to be optimal. We further analyze two existing triangular low-discrepancy point sets and prove that their mesh ratios are uniformly bounded, thereby establishing their quasi-uniformity. Finally, through a series of numerical experiments, we demonstrate that the proposed method provides an efficient and practical strategy for generating high-quality point sets on individual triangles.

Figures (8)

• Figure 1: Example of point sets generated by the VG algorithm from $n=3$ to $n=11$. In each subplot, the blue dots denote the generated points, the dashed lines denote the Voronoi edges, and the orange dots denote the candidate set.
• Figure 2: Mesh ratio of point sets generated by the VG algorithm for various isoperimetric quotients $J$, shown for $n=10$ (red), $n=20$ (orange), and $n=50$ (blue).
• Figure 3: Number of points required for the VG algorithm to achieve the optimal mesh ratio of $2$, comparing empirical results (orange) with theoretical bounds (blue).
• Figure 4: Six point sets with $n=45$: our VG algorithm (left top), barycentric grid (middle top), triangular van der Corput sequence (right top), triangular Kronecker lattice (left bottom), PD random (middle bottom), and i.i.d. random (right bottom).
• Figure 5: Mesh ratio of six point sets in the unit equilateral triangle: our VG algorithm (blue), barycentric grid (green), triangular van der Corput sequence (orange), triangular Kronecker lattice (red), PD random (black dash), and i.i.d. random (black).

Theorems & Definitions (35)

• Definition 2.1: quasi-uniform infinite sequence
• Definition 2.2: quasi-uniform sequence of point sets
• Definition 2.3: largest empty disk
• Lemma 2.4
• proof
• Definition 2.5: Voronoi diagram
• Remark 2.6
• Remark 2.7
• Lemma 2.8
• proof