Planarize voronoi curves on a revolved surface

Happy New Year!
I am trying to planarize a voronoi curves on revolved surface. I am not sure it is possible at all. I have found a script with kangaroo but the solution is not really good. I am not really familiar with Kangraoo but I could not make any better solution.
Is there any other way to get a better planarization solution?

voronoi planrize (21.7 KB)

Hi @Balazs,
Happy New Year

Planar tri-valent meshes are not possible on cylindrical surfaces.
On positively curved regions such as the round end of your shape it can work fairly simply by planarizing directly even with a fairly arbitrary starting mesh like this.

On negatively curved regions you can have planar polygons with tri-valent vertices if the polygons become concave. This requires significant changes to the geometry of the faces though, and often the mesh connectivity would also need to be changed.

In developable regions such as the cylindrical region near the base there’s no way the faces can be flat, tangent to the surface and with tri-valent vertices.

Consider for example a place where 3 faces meet

There is no possible way to cut these planes so that they meet edge to edge.
You would need to switch to a 4 valent mesh here to follow the curvature.

What is this for?

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Thanks for the detailed answer. Actually I am building 3d printed sculptures from flat building blocks. I have found interesting from the esthetical point view to build up (approximate) a regular shape (revolved surface) from irregular flat elements. The elements has to be planar because of the way of connections. Can you suggest an another approach to achieve this? (Maybe approximate the surface with quads and triangles?)

Can you tell us a bit more about the type of elements and connections you are thinking of?
If the panels don’t need to meet edge to edge, one option can be to just project each one to a local plane.
You can then still extrude them so they can be connected to each other at the sides.

You could make it with planar quads (either by designing a base quad mesh, or for a relatively simple shape like this it might work to use QuadRemesh as the starting point, then planarize), but it won’t have such an irregular look - the panels will need to follow the curvature directions of the surface, so it will look more grid like.

Triangles are the easiest option - they will always be planar.

Panels has to meet edge to edge. Is there a way to subdivide the surface into triangles and flat quads?