Ellipsoid mesh from curvature lines

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)