Constructive quasi-uniform sequences over triangles
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915916983304192 |
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| author | Xu, Hengjun Goda, Takashi |
| author_facet | Xu, Hengjun Goda, Takashi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructive quasi-uniform sequences over triangles Xu, Hengjun Goda, Takashi Numerical Analysis 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. |
| title | Constructive quasi-uniform sequences over triangles |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.07909 |