Hi !
I want to discretize a double curved surface into planar faces (but not triangular), using the method used in this paper :
https://www.researchgate.net/publication/300377122_Discretization_of_double_curved_surface
I try to reproduce this shape :
For the moment I have the intersection points of the tangent planes as shown below :
But I don’t know how to obtain a mesh made of planar faces like the example, does someone have an idea to solve this ? 
Thanks in advance, link to rhino and gh files :
Doublecurved.3dm (89.1 KB)
Doublecurved.gh (7.6 KB)
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Here’s a very simple implementation of the tangent plane intersection method
tangent_plane_intersect.gh (29.1 KB)
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Whow amazing this works perfectly ! Thank you so much !
Btw, is there a way to make this work even for surfaces which changes of curvature sign ?
It gets quite a bit more complicated when you have a mix of +/- curvature, and might need changes in mesh connectivity. It is possible though.
Personally I’ve always been a bit dubious about the tangent plane intersection method - although it guarantees exact planarity, all the implementations seem to result in rather awkward and oddly shaped polygons on general surfaces.
I’ve played more with planar hexagonal meshes by approaching it the other way around - starting from non-planar polygons and flattening them. Generating appropriate starting topology for that is also tricky for general surfaces though.
Would you say there is any benefit of using planar hexagonal mesh over a planar quad mesh? Or is it horses for courses depending on what type of surface is desired?
One nice thing about quads for beam structures is that you can locally align the 2 beam directions with the 2 principal stress directions, which is good for structural efficiency.
With hexagons you can only have 1 of the 3 directions aligned with stress, and I think the re-entrant/concave hexagons you get on anticlastic surfaces could be structurally problematic as beam layouts.
For plate based instead of beam based structures, 3 plates meeting around each node can’t fold without bending the plates, while 4 plates can form a mechanism like rigid origami (though generally a quad mesh as whole won’t be an origami mechanism unless you design it to be).
Quad panels will result in less wastage when cutting from rectangular sheet stock. I think cutting re-entrant hexagons from glass could also be tricky.
If you have a maximum of 3 edges meeting at a node (as in a hexagon grid), then getting offsets with torsion free nodes is easy by intersecting 3 planes. For quads where 4 meet at a node, the mesh has to be conical to get torsion free nodes, so you need a way to generate this conical mesh. This means you need to align with the curvature directions.
For hexagons though, unless you have simple synclastic shapes like domes, you also end up needing the layout to respect the curvature directions to get flat panels if you want them reasonably shaped.
So for synclastic dome-like structures where the plates form the main structure instead of the beams I can see some argument for hexagons. The rest of the time though quads have a lot of advantages.
This is all thinking about it in architecture, as node/beam or plate structures. For other structural systems or in different domains you could make different arguments for when one or the other is better suited.
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Thank you for all these explainations ! 
I really like the technique you have shared a few times where you make the quads planar and the mesh conical and end up with something like this…
I’m interested in whether the panels could be timber frame construction like this (only quads)…
Either cutting the dihedral (?) angles on the sides of the timber frame so the mesh faces remain planar (for glazing) or building the dihedral angle into each panel frame and cutting the top edges back to make the face planar again (which could require some pretty challenging compound angles in the panel corners).
Building the angles into the frame may allow for surfaces with greater curvature as too great a dihedral angle would cut the bottom of the timber frame edge.
But is this a beam or plate based structure? Each panel could be bolted to the next through the sides of each timber frame but this then makes each pair of adjacent panel edges a beam and the timber joints in the corner of each panel frame are what will dictate the strength of the whole structure I guess.
And so I assume that aligning the principal stress directions with the beam directions is still good for structural efficiency but the beams are not continuous across the structure since they are made up of segments relying on the timber panel frame corner joints for strength.
I guess I am trying to understand the limitations of making timber frames to build structures like this and whether this method of fabrication is going to require some careful design constraints when designing the structure form (i.e. limits of curvature, mix of +/- curvature, quad / tri / hexagon, etc etc)
