Double-curved Paper Models – Double-Layer Diagrid Unfold Method


One of my interest in computational geometry since the ‘Zip’ project is the research to realize complexity. I’m dealing with double curved geometry recently, which is famous for being very difficult to be fabricated.

My research here is the prototyping process of double curved geometry. Previously, I have seen a successful attempt by Axel Kilian (my tutor some time ago), in his paper in MIT in 2003 described a process: Fabrication of partially double-curved surfaces out of flat sheet material through a 3d puzzle approach, which is concerned about the prototyping of double curved surface using only a laser cutter and flat sheet materials to realize double curved surface.

The method I present here is to utilize a laser cutter and flat sheet paper, to produce double curved models. The adventage is the convenience in assembling the model and the flexibility of the input.

Description of the computation logic

The computation starts with dividing the geometry into a diagrid pattern. The idea of the division is to approximate the double curved surface by locale planar (or developable) surface, although other patterns (such as triangular grid) could be used, the diagrid pattern is the best subdivision approach for the subsequence logic and assembly, having a high degree of accuracy by utilizing the property of the paper to bend into single curved surface. Because we can treat each quad as two triangles (and because triangles are always planar) , we can abstract the curved surface of each quad into two triangles which is subsequently assembled by a continuous sheet of paper.

This is why quads are more preferred than triangles, because within a fixed subdivision density (which is likely the result of the paper material failing to perform), more quads are subdivided than two times of their triangles. A finer subdivision will obviously improve the approximation, so quads are better in that terms. The only draw back is that in order for the paper to bend, a slightly stronger glue (and more generous portion) have to be used (explained later).

Prior art

The core idea of the script is the assembly method. Imagine we are able to figure out the unrolled geometry of each quad, layout and laser cut them, it will be a nightmare for anyone to assemble them back to a surface. First problem is that there are a lot of surfaces, like a giant puzzle. Second problem is that you have to provide a tab (which is an extra bit of paper sticking out of the quad) for gluing purpose, the tab have to be big enough for fingers to manipulate while trying to avoid dominating the surface.

Prior art such as this one are easy to produce, but difficult to assemble.

We can foresee that this method will create a very bumpy and messy side while only one side could be kept relatively smooth. Another problem is that the model quality dramatically lowered because of imprecise folding of the tabs. The folding behaviour of paper (and general thin cardboard) depends on the depth of the laser score; shallow score will produce wrinkled and curvy fold; deep score are better because they will produce a cleaner edge, but this will expose the upper and middle layers of the paper at the ~90 degrees fold, and adds some distance to the hypothetical joint where material thickness are not taken into consideration. It is unavoidable that tabs will add an extra distance between two panels at the joint, which will distort the global geometry when the whole surface is assembled.

Deep scores are also difficult to archive with the laser cutter, this is due to inconstancies in paper thickness (on different area of the paper), inconstancy in laser power at different location, inconstancy in laser power when the XY mirror move in different angle, and the fact that thin paper tends not to lie flat on the cutter bed, affecting the laser focus. This is why a high power laser setting attempting to create a deep score might risk cutting completely through in certain area of the paper.

Folding and gluing the tabs are also tricky and imprecise because the tabs will add an extra distance between two piece. No matter how careful you fold and glue, there is a gap between two panels (the upper and middle layers of the paper exposed by the score and fold method). Because the tabs have to be folded almost 90 degrees, the gap and the tendency to  which is unavoidable.

New assembly method – using spray adhesive (spray mount)


Model Photos:

Test1 – Snake

Regular offset on grid to generate a ‘frame looking’ surface. I founded out that the void in each quad help align the two layers of surfaces. and make the paper more flexible during the gluing.

The gaps between strips are minimized as much as possible, it took some practice to do it right.

Test2 – Torus

Complete loop in both U and V direction of the surface, the strips are able to end at different location, so as to provide a key for finishing the loop.

The void/openings are not a regualr offset but a parametrically modelled geometry, it varied in size depending on a number assigned to each of the quads, in this case the numbers came back from Ecotect solar analysis.


Test3 – Intersection

An intersection is difficult to fabricate because it involves two trimmed surface. Trimmed surface are tackled here by first ignoring the trim line and create the cut files from the untrimmed surface; the trim lines (boundary) are mapped from the curved surface to the flattened cut file afterwards. The trim lines are cut with reduced laser power so that each strip remain more or less intact, this will help reconstructing the puzzle. The trim lines are completely trimmed off with knife (pretty easy) during assembly.

Two ribs are made to strengthen the edge of the boundary. They are a strip of ruled surface which can be deprived from the two original surfaces, by offsetting the boundary line on the surface.

If glued precisely enough, the rib should match the boundary of the surface model and the two rib should now match. Note that the two ribs are not identical, not even similar, because although they met at the same line, they follow two different surface.

Test 4 – Just make some more


Shadows – 🙂