[emmanueloga_]

One of my favorite renderings of this is in this article [1], that shows the relationship between the Delaunay triangulation, Voronoi diagram, a relative neighborhood graph (RNG) and the euclidean minimum spanning tree (EMSP) of a set of 2d points.

I have a theory that the union of the edges of the RNG and EMSP could be used for automatic navigation between widgets in a GUI: combining the two, there's always between 1 and 4 edges coming out of every point, and so each of them could correspond to a direction key up/down/left/right according to some simple heuristic.

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1: https://axltnnr.io/2018/Triangulation/

[kian]

This is pure speculation (graphics algorithms and geometry aren't my strengths), but if you were to do this with a set of 3d points, would we get a relationship between delaunay triangulation (pyramidalization / convex hull?) a relative neighborhood graph, and a minimum spanning tree of 3d points, where combining the RNG and the MST would yield between 1 and 8 edges coming out? (at which point, we could use, say, the numpad for 3d gui widget navigation in the meta-verse in the same way you suggest using 4 cardinal directions for 2d gui representations)

[emmanueloga_]

Interesting question! By virtue of being a tree, the MST produces at most 3 edges coming out of any vertex, so this should be the same in 3D. The MST then adds (sometimes) a 4th edge, so, although you could build both graphs in 3D space, you would still end up with 4 edges coming out of any vertex, I think.

In 3D space the Delaunay triangulation would produce a bunch of irregular tetrahedra, so the edges coming out from every vertex would vary between a minimum of 3, and a maximum of 12, if I get it right (ref: [1] :-).

The 3D Voronoi cells are another story... I found some implementation that you can play with to see how it looks [2] [3], each cell is of a shape called "convex polytope". It feels like these cells are packed like each of the sub-cubes of a rubik, but I'm not 100% sure :-) ... if that's true, you could jump from each vertex to the next in at most 26 directions? (hand-waves :-p)

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1: https://en.wikipedia.org/wiki/Tetrahedron#/media/File:M_tic....

2: https://github.com/BrunoLevy/geogram/wiki/Delaunay3D

3: https://math.lbl.gov/voro++/examples/

[eutectic]

I think convex hull is also closely related.

[OskarS]

Indeed: the convex hull is the outer boundary of the Delaunay triangulation. In addition: if you "bow out" the points into 3D, the 3D convex hull is exactly the Delaunay triangulation. These two concepts are very closely related.

[hyperific]

Super cool resource! Thanks for sharing this!