Since the thickened curve meshes are all quads, with all even valence vertices, they can be sorted by direction and turned into strips

Very nice.

It could also be useful to make flat quads by moving one edge each 2 edges like here

or

Cool! I built a similar workflow through Kangaroo as well, as an exercise in mesh modelling and topology for students last November:

@laurent_delrieu, nice work, I developed a pleating strategy for strips too some time ago (https://www.instagram.com/p/BpwkadBhGbk/) but I never connected it directly with quad planarity. Can you please explain what geometric principle ensures planarity in your pleated example? Just moving alternate edges shouldnâ€™t be enoughâ€¦ or am I missing something?

Nice!

I think what Laurent is showing and Marcâ€™s work ensures planarity through a geometric construction creasing each strip into a ridge.

Iâ€™m guessing by intersecting rotated planes through either side of the input strip to find a line for the crease that is coplanar with both. Is that along the right lines @laurent_delrieu ?

Itâ€™s a nice technique because as well as guaranteeing planarity so the strips are developable, the pleating improves the structural rigidity.

It would also be possible to make the strips developable while keeping the surface smooth, by optimising the vertex positions. Iâ€™ve shown this before for strips which are aligned with the pipe directions, but I think it could also be done with diagonal strips like this - if the mesh was altered by connecting adjacent diagonals, so that the faces still followed the curvature directions.

So instead of this (which would be impossible to planarize)

youâ€™d have this:

@ale2x72 @DanielPiker , Daniel yes it is as you described for sure it is useful to augment the rigidity and ease the fabrication by lowering the number of folding lines.

I have developped a function to make the crease. So you specify an angle and an offset angle, the algorithm begin at the beginning of one strip. and try to go the end without being sure to end with the same angle as the the beginning. So at the moments it lacks optimization. If one day I want to use it I will have to make some optimization.

I interpreted your screenshots and made some planarization tests with Kangaroo by altering the mesh topology (connecting adjacent diagonals).

original mesh, â€śplanarizedâ€ť

altered mesh, planarized

altered mesh, planarized, other stripping direction

Iâ€™ll try some laser cutting in the next days!

Thanks everyone for the nice inspiration!

Marco