I have some planarization issues. My task is it, to planarize the subsurfaces of some tessellations. (It should look as much as possible like the original tessellation at the end). The first one is a simple quad pattern. This works awesome with my definition. The second one is a hex-pattern. This works partly, but not how I image it to. And the last one is a voronoi pattern. This causes an error from my kangaroo solver and I dont have a clue why. Does anyone had the same issues? Or does anyone find my fault?

I would be pretty surprised if you manage to find something meaningful for those initial patterns on an anticlastic surface, as the planar segments will look very different from the starting pattern.

I agree.
Planarization like this usually only works well when your initial panels are at least somewhat close to a good planar layout.
So for quad meshes, this means starting out at least roughly in the direction of principal curvatures.
It can also work for hexagonal meshes on a positively curved surface, such as a dome.
This way the points only have to move a small distance, so the mesh stays intact, especially if we use some other goals to keep the panels fairly regular.

If you have nodes with only 3 edges attached, like in your hexagon and Voronoi options, the only way the panels will be able to form a region of negative Gaussian curvature is if the panels become concave.

Starting from some arbitrary polygonal mesh, the points will have to move a large distance, completely changing the shape of the panels, with some edges perhaps even disappearing, so we might need some topology changes to maintain a good mesh.

So do I understand it right? Either the surface is changing oder the panels are changing while in each case the other component is staying nearly like it was before?

is Kangaroo able to perform the topology changes? I’m very interested in the panels, even if there are some edges disapearing or the panels changing their shapes?

The generation of PH-Meshes is a quite large research field, it’s a bit over the top to sum those up here (there are many publications on that issue, and also quite a couple of forum posts I would guess).

In your specific case with a anticlastic surface the panels will become concave, as Daniel Piker already pointed out (They will look like bow ties / hour glasses…). The voronoi pattern and patterns created from planar regular hexagons mapped on a surface are convex and thus look very different (and are also potentially very curved in the beginning, so they will have to move a lot to achieve a planar configuration). Going from one to the other will most likely be very problematic for any kind of constraint solver, particularly if topology changes are occuring, which I guess is likely in the case of Voronoi Patterns, as their topology tends to be quite arbitrary if the seeds are placed randomly on a surface.

If I understand it correctly a topology change would mean a hexagon suddenly becomes a pentagon or a heptagon. So the amount of points inside the planarity goal would have to change during the solution (I think ?). I am not sure if and how this is possible inside Kangaroo.

So I need a third tesselation that I can planarize for my thesis. I got quads and I got hexagons. triangles are in fact the same as hexagons or quads. Has anyone an idea?