## Abstract

In this paper, we present a simple yet efficient algorithm for triangulating a 2D input domain containing a Poisson-disk sampled point set. The proposed algorithm combines a regular grid and a discrete clustering approach to speedup the triangulation. Moreover, our triangulation algorithm is flexible and performs well on more general point sets such as adaptive, non-maximal Poisson-disk sets. The experimental results demonstrate that our algorithm is robust for a wide range of input domains and achieves significant performance improvement compared to the current state-of-the-art approaches.

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## Acknowledgments

This research was partially funded by National Natural Science Foundation of China (Nos. 61372168, 61172104, 61331018, and 61271431), the KAUST Visual Computing Center, and the National Science Foundation.

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## Appendices

### Appendix A: Proof of correctness

In this appendix, we prove that our triangulation algorithm correctly computes a complete triangular mesh. Here, the collection of the cells that belong to one sample \(p\) is called the Voronoi region of \(p\).

###
**Property 1**

Each Voronoi region generated by grid cell clustering is connected and any two regions are non-overlapping.

###
*Proof*

In the clustering step, we adopt a propagation algorithm equipped with a priority queue on the grid to approximate the Voronoi regions. Each time we choose a grid cell that has the highest priority and assign the label of its nearest sample to it. Note that each grid cell is connected to a neighbor cell having the same label, which is in turn inductively connected to its nearest sample. In addition, each grid cell will be assigned only once and never changes its label. As a result, the propagation algorithm ensures that each Voronoi region is singly connected and any two regions are not overlapping.

###
**Property 2**

No two triangles in the triangulation intersect each other.

###
*Proof*

Suppose that a triangle \(\triangle abc\) crosses another triangle \(\triangle def\). There are two configurations of the intersection: the two triangles do not share common vertices (Fig. 13a) and share common vertices (Fig. 13b, we take the 1 common vertex as example, the proof for the other cases is straightforward). The former configuration is impossible; otherwise, the Voronoi region of \(d\) must connect to the Voronoi regions of \(a\), \(b\) and \(c\), and this will generate triangles \(\triangle abd\) and \(\triangle adc\) instead of \(\triangle abc\). The latter configuration happens only if some Voronoi regions are not connected, such as the green cells shown in Fig. 13c. This contradicts the fact that all the Voronoi regions are connected (Property 1). Thus, no two triangles in the triangulation intersect each other.

###
**Property 3**

The union of all triangles fully covers the input domain. In other words, there are no holes in the triangulation.

###
*Proof*

Firstly, our boundary sampling and clustering steps guarantee that for any two consecutive points on the boundary there will be an edge between them. This ensures the output triangulation is tightly confined to the input domain.

Next, we demonstrate that any holes in the interior of the domain will be filled by triangles. Before that, we note all the grid cells have been assigned a label (Property 1). Now suppose there exists a hole, which is a polygon consisting of three or more vertices. We consider the Voronoi regions of any three consecutive vertices \(a\), \(b\), \(c\) in counterclockwise order. There are only two cases. First, if the three Voronoi regions connect to each other, then our algorithm will generate a triangle \(\triangle abc\) and the hole shrinks to a smaller one. Second, if the Voronoi regions of \(a\) and \(c\) are not adjacent, but they both connect to that of \(b\), then there must be another vertex \(d\) whose Voronoi region connects to all of the three Voronoi regions. In such a case, it will generate two triangles \(\triangle abd\) and \(\triangle bcd\) and the hole also shrinks to a smaller one. As such, the hole becomes smaller and smaller by repeatedly processing like this, and finally it will disappear.

### Appendix B: Input domains

See Fig. 14.

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Guo, J., Yan, DM., Bao, G. *et al.* Efficient triangulation of Poisson-disk sampled point sets.
*Vis Comput* **30, **773–785 (2014). https://doi.org/10.1007/s00371-014-0948-z

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### Keywords

- Triangulation
- Poisson-disk sampling
- Geometric algorithms