Thank you it is super nice, and I can get what I needed:)

Do these conical qualities are also valid for hexagonal meshes?

The simplest ones.

Thank you it is super nice, and I can get what I needed:)

Do these conical qualities are also valid for hexagonal meshes?

The simplest ones.

I believe *any* planar hexagonal mesh with 3-valent vertices is conical.

See https://www.microsoft.com/en-us/research/wp-content/uploads/2016/12/A-Note-on-Planar-Hexagonal-Meshes.pdf

I believe mobius transform cannot maintain the planarity of hex meshes as in quads?

The only way is to go through relaxation (for the case of simple sphere)?

After simple planarization (projection to plane 300 iterations) I used Mesh.Offset method from Rhino.Common to offset mesh in both directions.

The lofting edges does not produces planar extrusion.

I attached the grasshopper file with transformation.

Planarization.gh (34.2 KB)

Question:

Is there specific method to calculate the vertex normal to offset mesh edges that will remain planar?

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Möbius transformations will only preserve planar hexagons if their vertices start out on a common circle - so only the regular hexagonal grid or other Möbius transformations of it (as shown in this post).

The correct vertex normal to use for the offset to ensure planar beams is the intersection of the bisecting planes between each of the surrounding faces.

(Actually this is true for V-V offsets of quad conical meshes too - I think maybe it was only a coincidence that it was working with the standard offset in the example I posted above)

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