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.
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)
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)
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.
Comments
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/
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)
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/
I think convex hull is also closely related.
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.
Super cool resource! Thanks for sharing this!