# How can one prove the duality of Voronoi and Delaunay?

Hoping I'm not misunderstanding the concept here, but it is my understanding that Voronoi Diagrams and Delaunay Tesselations are 'dual' to one another, owing to the fact that each' solution makes computational of the other's solution trivial.

What I'm looking to understand is, why? I mean, it obviously happens, but is there some kind of mathematical proof or principle that justifies that convenient coincidence? Or is it just that, a coincidence?

I suppose, the voronoi quality of 'each vertex is equidistant from the 3 closest points' means that any matching triangulation is a delaunay triangulation (there is no point closer than the closest points), but I am hoping for something more... in-depth? Is there such a thing?

To clarify: the question is not about what them being dual means, like the suggested duplicate. The question is why they behave as duals.

• May 7, 2019 at 12:57
• Check the last answer in the link provided by GoHokies: en.wikipedia.org/wiki/Dual_graph?wprov=sfla1 May 7, 2019 at 13:19
• @GoHokies that question has a key difference to mine: that one is about how duality is applied for the two graphs. Mine is about how it emerges, or 'why' it is applied. May 7, 2019 at 15:41
• @nicoguaro that answer explains how to use the duality to go from one solution to the other. I am hoping for a mechanism to explain why this works in the first place. I'll look into that link when I can focus properly, though, it might help. Not sure why I missed it. May 7, 2019 at 15:44