chouette ! ça prend tournure ! ;)

Drawing of a big number of pieces are not so evident! :)

Maybe with an unfolding...from an existing surface...

To continue the exploration! :)
Maybe the Flow can be made in one pass...against two here
but as it's so simple like this...

Just to find the Formula of the draft "Angle"

Of course here the Start surface has all planar surfaces!

It appears that drawing a losange between each middle of a no planar rectangle
give a planar surface!



Then with intersection that gives the "rectangle planar" !
and verified by the planar function!


Alas if all rectangles are planar they are not aligned between them!
After it's a big headache maze to make each facet rectangular "jointive"!

After some consuming time! :)
Now all yellow facetts are planar with some functions like Orient Line/line...Trim...Scale etc...



Diagonal alignment is like an optic effect! :)

Problem is to make the alignment between them of numerous facets!

Maybe a trick is existing! :)

The best if of course to make from the begining directly a generative structure of "planar facets"
for avoid all this hand adjustment headache!

Who is not reasonable times consuming!

And here there are just 16 facets...in normal case that is more like 200 - 300! :)

Sorry the file with full alignment facets was not saved in 3dm format (so a good exercice! :)
only in SKP format! :(

https://moiscript.weebly.com/uploads/3/9/3/8/3938813/milieu.3dm (missing top alignment)

https://moiscript.weebly.com/uploads/3/9/3/8/3938813/end_precision.skp (full alignment)

A funny thing!

Each Red quad's struture is not "planar...

But losanges (from the middle of the quads as shown above are planar!

If you draw from each middle of a losange a closed polyline so a quad : this one will be also planar!

Triangles resultant are of course "planar" :)

So all elements of the surface under the Red structure is now planar!

A resultant problem for end holidays :)

Does always existing a Point P (x,y,z ) for make 4 planes P1, P2, P3, P4
from vertices A,B,C,D,E,F,G,H given on the 3D space ?

And how to find it ? ;)
https://moiscript.weebly.com/uploads/3/9/3/8/3938813/4_plans.3dm

If point P exists, there would be 4 planar faces, P1...P4.
According to the link:

https://blender.stackexchange.com/questions/46113/how-to-make-all-quads-or-ngons-on-your-mesh-planar-2d

"when a vertex has more than 3 surrounding faces, their planar intersection does not necessarily exist - it would be actually very rate if it did:"

- Brian

But we have intuition that is maybe possible! :)

Point

Sorry i can't Planar this one in your file!

I think it's not possible with your file.

P1, P2, P3, P4 are defined by 3pts for each as all planes 3 points define a plane.
P1 (A, B, H)
P2 (B, C, D)
P3 (D, E, F)
P4 (F, G, H)


The question is : does exist a point P which is on P1, P2, P3 and P4 ?

I think, we have two possibilities to determine it :

1/ Mathematical with a system of equations (each plane is determined by a equation) and its resolution.

2/ MOI (or CAD drawing)
if we do planes P1, P2, P3 and P4 with points A,B,C,D,E,F,G,H with their 3 points, we can do intersections of these 4 planes. If we have only a point... we win and have P !
With your file, the result is a small thetahedron... P does not exist.

My analysis. right or wrong ?

I suppose yes because all edges A to H are "rigid" and that limits variation but in some cases we can make it...

I am waiting answer from a guy who has found a solution but i believe there is an artefact!

@ Rudl

With your file i can make it if i Up "E" on "Z" vertical by 0.05 unity! ;)
I will verify if i obtain the same with the original...



Ok I obtain the same result with the original... 0.05 unity :)

Maybe this distance can be reduced by taking another orientation than the Z vertical line?...

In any case for the moment this point P seems impossible with this given structure!

@Phiro
You seems right...the answer that i was waiting confirm your...there was a similar construction like mine above...

Point P with this structure seems not possible...very tiny but not..we must move slightly point E !
(so some others not shown here) :)

Similar project with same workflow for a dome I did a while back.
I used RHINO UNROLL Command to flatten all faces.

3D world is a small planet! ;)

We have to be allowed to dice it up into more than 4 surfaces....

Ah that is a brilliant answer! Bravo...

You have the original structure Yes!
You fill the hole Yes!
You have only planes Yes!
You have not 4 planes but 8 so better twice! :)
And you have only quads Yes!

And there is an infinity of this solution! Excellent!
Burmann's Triangles wins ! ;) Bermuda Triangle ? :)

In situ



More flat


Complete



Now the best will be a little script or Elephant for automatise the down levels
or maybe a tricky use of existing functions (Loft, Sweep, NetWork...) for have directly plane surfaces from false surfaces! (dream :)
I have seen during this little exercise than NetWork has numerous parameters! (Uniform with number of Points...)

By hands it's 20 seconds for the last 4 top quads ;)

For this : here by hands its nothing for 12 quads but when you have 200-300 it's another story :)

It's true Quads "plane" built by external perimeter!


Not like that but...