Is there a function to compute the area of the triangle that include any point in a 2D points set?

Hello, everyone.

I am an engineer, and I want to know the way to assess the evenness of the distribution of points in a 2D point set.

And my idea is that:

• For any point P, there must be a triangle that including the point P in it, and the triangle is the particular triangle has the minimum area among the triangles that including point P in it.
• Get all the triangles area and compute the variance of the areas of the triangles.

But, find the triangles with the minimum area including point P and compute the area is an unfamiliar work for me.

So, anyone know is there has a function has this functionality?

If anyone knows, Please tell me!

[John D'Errico]

Huh? Your idea is wildly confusing. No, there is certainly no way to do something that you cannot even explain.

At the same time, there are surely many ways to describe how uniformly scattered a set of points may be in the x-y plane. You might start by looking at the nearest neighbor for any point.

Answer 1

[Jan]

This cannot work. If it works, it would work for a bunch of point placed on a circle. But how do you want to define the triangles in this case?

You can divide the area into polygones, see https://en.wikipedia.org/wiki/Voronoi_diagram. This is related to the Delauny triangulation, because the edges go perpendicular through the midpoints of the triangle sides. But as you see, the Voronoi distribution is not built by triangles in the general case.

Even the Delauny triangularisation is not unique. Imagine 4 points: 3 are on the corners of a square, one has a larger distance to the center. Then you can either create two equal triangles (which would mean an "even" distribution in your idea) or two triangles with different areas. So which one is the right solution?

The points on the border cannot have a surrounding area.

[The Pirate]

Thanks for your answer, Jan Simon, especially your link to wiki encyclopedia.

It is right to use polygon rather than triangles to divide the point set, and you let me know that the problem is the so-called Voronoi diagram problem!

in fact, I search on the internet and find an algorithm that does this work: Voronoi diagram in a circle, that is exactly what I want. This is the Link: circle bounded Voronoi diagram

Thank you!

Answer 2

[Image Analyst]

Alex:

What you want is "spatial statistics" and there is a lot of work in that field. The best book I have found is by Adrian Baddeley at CSIRO (Australia) There are lots of methods over 171 pages for characterizing point patterns into degrees of "regular/periodic", "random/Poisson", and "clustered". Check it out. It's a fascinating book.

http://umaine.edu/computingcoursesonline/files/2011/07/SpatstatmodelingWorkshop.pdf

First go down to section 1.2.2 on page 7 and see the patterns. Then you might go down to page 53 and look at "Tests of Complete Spatial Randomness". Good luck.

By the way, there is a function polyarea() that will get you the area of a triangle or any polygon.

[The Pirate]

Thanks for your answer, Image Analyst !

I have found that the book you recommended is very instructive. Since I am not familiar with this field, I need to work on this book, and I believe that the book will be beneficial to my future work.

Thanks.