<p>If it is a regular polygon its barycentre <span class="math-container">$c$</span> can be found by calculating <span class="math-container">$$c = \frac{1}{n}\sum_{i=0}^n v_i.$$</span> Then you can construct <span class="math-container">$n$</span> triangles <span class="math-container">$c,v_i,v_{i+1}$</span> which can be regularly sampled like the triangles you showed.</p>

CC BY-SA 4.0
2024-10-28T13:53:23.433Z
2024-10-28T13:53:23.433Z
Reynolds
<p>There is a sampling pattern used in FEM method for triangles, which can be seen here:</p>
<p><a href="https://i.sstatic.net/jy8qewlF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jy8qewlF.png" alt="enter image description here" /></a></p>
<p>The rough description is that at each edge we have n samples (containing the endpoints) and then in the interior the number of samples remains roughly uniform.</p>
<p>The equivalent for the square is just a regular grid of points, like this one:</p>
<p><a href="https://i.sstatic.net/nSsxxkEP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/nSsxxkEP.png" alt="enter image description here" /></a></p>
<p>What I am looking for is, given a number <span class="math-container">$n$</span> and a regular polygon with <span class="math-container">$n$</span> sides, a rule such that:</p>
<p>Each side of the polygon has <span class="math-container">$n$</span> samples (including the endpoints), the interior is regularly sampled (i.e. the minimum distance between points is bounded).</p>
<p>I am not having much luck finding out how to do this.</p>

<mathematics><computational-geometry><geometry>
Generalized Sampling pattern for regular polygons?
Makogan
# Generalized Sampling pattern for regular polygons?

(Asked by: username_0 on 2024-10-26)

There is a sampling pattern used in FEM method for triangles, which can be seen here:

[![enter image description here](https://i.sstatic.net/jy8qewlF.png)](https://i.sstatic.net/jy8qewlF.png)

The rough description is that at each edge we have n samples (containing the endpoints) and then in the interior the number of samples remains roughly uniform.

The equivalent for the square is just a regular grid of points, like this one:

[![enter image description here](https://i.sstatic.net/nSsxxkEP.png)](https://i.sstatic.net/nSsxxkEP.png)

What I am looking for is, given a number $n$ and a regular polygon with $n$ sides, a rule such that:

Each side of the polygon has $n$ samples (including the endpoints), the interior is regularly sampled (i.e. the minimum distance between points is bounded).

I am not having much luck finding out how to do this.

**Tags:** mathematics, computational-geometry, geometry


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## 1 answers

### Answer by username_1 on 2024-10-28. Score: 2 | Accepted answer

If it is a regular polygon its barycentre $c$ can be found by calculating $$c = \frac{1}{n}\sum\_{i=0}^n v\_i.$$ Then you can construct $n$ triangles $c,v\_i,v\_{i+1}$ which can be regularly sampled like the triangles you showed.


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