<p>This animation from the wikipedia article on delaunay triangulations shows you an example of where the delaunay triangulation will switch from having a horizontal interior edge to a vertical interior edge:</p>

<p><a href="https://upload.wikimedia.org/wikipedia/commons/8/82/Delaunay_triangulation_does_not_minimize_edge_length.gif" rel="nofollow noreferrer">https://upload.wikimedia.org/wikipedia/commons/8/82/Delaunay_triangulation_does_not_minimize_edge_length.gif</a></p>

<p>The moment this configuration switches to a horizontal interior edge, it is showing that the delaunay triangulation prefers a triangulation with a smallest maximum angle of 88-ish degrees to one that has a smallest maximum of 66-ish degrees.</p>

CC BY-SA 4.0
2020-03-11T10:36:05.897Z
Thank You. I think you mean 120-ish instead of 66-ish?
Inuyasha Yagami
CC BY-SA 4.0
2020-03-15T03:46:36.653Z
I meant 66ish because I thought we were considering the minimum (smallest) of all the maximum angles in the triangulation, where a 'maximum angle' is the largest angle in each triangle. So I was looking at the smallest of those maximum angles as our objective value. But if you just mean to minimize the overall largest angle in the whole triangulation, then, yes, 120-ish. Luckily, the transition in the example gives an example for both measures.
JimN
CC BY-SA 4.0
2020-03-11T02:06:04.123Z
2020-03-11T02:06:04.123Z
JimN
<p>I know that that the Delaunay triangulation maximizes the minimum angle of triangulation. 
And it does not minimize the maximum angle. If we consider the set of points in <em>general position</em>(no four points are co-circular). Can someone please give a simple example where the Delaunay triangulation does not minimize the maximum angle.</p>

<computational-geometry><delaunay-triangulation><voronoi>
Example of Delaunay Triangulation where it does not minimize the maximum angle
Inuyasha Yagami
# Example of Delaunay Triangulation where it does not minimize the maximum angle

(Asked by: username_0 on 2020-03-10)

I know that that the Delaunay triangulation maximizes the minimum angle of triangulation.
And it does not minimize the maximum angle. If we consider the set of points in *general position*(no four points are co-circular). Can someone please give a simple example where the Delaunay triangulation does not minimize the maximum angle.

**Tags:** computational-geometry, delaunay-triangulation, voronoi


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## 1 answers

### Answer by username_1 on 2020-03-11. Score: 2 | Accepted answer

This animation from the wikipedia article on delaunay triangulations shows you an example of where the delaunay triangulation will switch from having a horizontal interior edge to a vertical interior edge:

<https://upload.wikimedia.org/wikipedia/commons/8/82/Delaunay_triangulation_does_not_minimize_edge_length.gif>

The moment this configuration switches to a horizontal interior edge, it is showing that the delaunay triangulation prefers a triangulation with a smallest maximum angle of 88-ish degrees to one that has a smallest maximum of 66-ish degrees.

> **Comments:**
> **username_0**: Thank You. I think you mean 120-ish instead of 66-ish?  
> **username_1**: I meant 66ish because I thought we were considering the minimum (smallest) of all the maximum angles in the triangulation, where a 'maximum angle' is the largest angle in each triangle. So I was looking at the smallest of those maximum angles as our objective value. But if you just mean to minimize the overall largest angle in the whole triangulation, then, yes, 120-ish. Luckily, the transition in the example gives an example for both measures.  

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