# Methods for Creating Curved Shell Structures From Sheet Materials

## Abstract

**:**

## 1. Introduction

## 2. Previous Work

**Figure 1.**Two ‘S’ shaped splines constructed using edges of three connected planar quadrilaterals. Note that the two splines are not in the same plane and indeed that they need not even be planar.

**Figure 2.**Attempting to create a developable surface from two ‘S’ shaped splines above, this surface can only be made developable by introducing the radial singularity shown at the top-middle.

**Figure 3.**‘C’ and ‘O’ shaped simple developable surfaces and their construction. The red and green quadrilaterals are each planar, however none of the blue quadrilaterals need to be planar in order to generate developable surfaces.

**Figure 4.**‘C’ and ‘O’ shaped simple developable surfaces unrolled. Note that the ruling lines are irregular. They are neither radial nor parallel, but change direction constantly along the length.

## 3. Triangles and Quadrilaterals

- Consecutive creases share a common intersection—this is the basis of a ‘conical’ roll.
- Consecutive creases are parallel—this is the basis of a ‘cylindrical’ roll.

## 4. Method One

- This quadrilateral is planar.
- This quadrilateral has two parallel edges.
- This quadrilateral has two edges that intersect at the origin of the scale transformation.

**Figure 5.**Two parallel homothetic lines (red), scale origin, two radial edges (green) and resulting planar quadrilateral.

**Figure 6.**Parallel homothetic polylines (red), scale origin, radial edges (green) and resulting three planar quadrilaterals.

- All quadrilaterals are planar.
- Each quadrilateral has two parallel edges.
- All remaining edges intersect at the origin of the scale transformation.
- This is a developable surface.

_{0}, O

_{1}and O

_{2}of the three scale transformations form a straight line. This is important later when creating secondary shapes based on more than three primary surfaces where the origin-points need to be co-linear in order to generate planar crease lines, see method three below.

**Figure 10.**Unrolled developed cutting pattern for rigid tube shown in Figure 9.

## 5. Method Two

**Figure 14.**Developed surfaces created using method two, note that the ends of each panel are parallel.

## 6. Method Three

**Figure 15.**Homothetic splines (magenta) and their regular division into secondary geometry (blue). These create a secondary set of splines (green).

## 7. Method Four

**Figure 18.**A primary geometry as created in method two, see Figure 13, with homothetic lines from the surface (red) used as construction for a secondary set of homothetic splines.

**Figure 19.**Developable surfaces generated by ruling between the secondary homothetic splines already created, see Figure 18.

**Figure 20.**Developed surface created with method four. This is the 3D shape shown in Figure 19.

## 8. Method Five

**Figure 21.**Four reflection mapped splines, three mirror planes, and the generated developable surfaces.

## 9. Homography

_{1}, y

_{1}, z

_{1}component then needs to be divided by w

_{1}:

**Figure 23.**Developed surfaces before and after a perspective homography transformation. The rear shape was created using method three. The transformed developable surface at the front is not possible to create directly through any of the five methods discussed previously.

**Figure 24.**Developed surfaces before and after a perspective homography transformation. The outline on the right is the untransformed shape developed, and the outline on the left is the developed version of the transformed shape.

## 10. Built Work

**Figure 27.**Assembly of one segment. This is 18 m long and the full sculpture consists of eleven different segments.

**Figure 28.**Construction. The segments are installed incrementally. This photo shows the sculpture approximately 80% complete. The props provide temporary stability during construction and will be removed on completion. Photo by S. Horrod.

**Figure 29.**Final shape. The black attachments are mounting points for the wheels of a display car. Photo by S. Horrod.

**Figure 30.**Completed structure. This is a photograph. Cars are classic Lotus Formula 1 cars from 1964 to the present day.

## 11. Speculative Works

**Figure 34.**Unrolled panels for Figure 33.

**Figure 36.**Unrolled panels for Figure 35.

**Figure 38.**Unrolled panels for Figure 37.

**Figure 40.**Unrolled panels for Figure 39.

**Figure 42.**An unrolled method four surface from Figure 41.

**Figure 43.**Multiple method one surfaces in a free-form shape. This shape has an additional homography transformation.

**Figure 44.**Unrolled method one surfaces from Figure 43.

## 12. Materials and Practical Considerations

## 13. Additional Uses

- Concrete shuttering. This can be either permanent steel shuttering providing reinforcement for the concrete inside, or temporary plywood for more traditional concrete systems with internal reinforcement.
- Internal steel reinforcement for concrete shuttered as described above. This would be made in the same way, but with perforations and tags to key into the mass concrete and/or to support curved reinforcement bar.
- Hollow cores for concrete shuttered and reinforced as described above.
- Ductwork, curved spline shapes can be used to traverse awkward and confined spaces. Cold rolled-edge standing-seam techniques can be used to assemble long span self-supporting ductwork on-site.
- Linear structural frameworks supporting walls, floors, roofs, glazing, or cladding.
- Being able to reliably construct arbitrary 3D splines from plate materials should be useful for track supports in amusement park rides.
- Self-supporting shell and bulkhead stiffened shell structures as shown in method three and method four will be lightweight, strong and relatively straightforward to construct.
- Planar glazing and cladding systems. Any shape that can be constructed from rolled developable surfaces can also be constructed from quadrilateral planes of glass or rigid panels with steel framework following the edges, although in this case, strictly planar curves will be more useful for generation of planar steel frameworks.

## 14. Further Work

## Acknowledgments

## References

- Shelden, D.R. Digital Surface Representation and the Constructibility of Gehry’s Architecture.
- Kilian, M.; Fl¨or, S.; Chen, Z.; Mitra, N.J.; Sheffer, A.; Pottmann, H. Curved Folding. ACM Trans. Graphic.
**2008**, 27, 75:1–75:9. [Google Scholar] - Liu, Y.; Pottmann, H.; Wallner, J.; Yang, Y.; Wang, W. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph.
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© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Postle, B.
Methods for Creating Curved Shell Structures From Sheet Materials. *Buildings* **2012**, *2*, 424-455.
https://doi.org/10.3390/buildings2040424

**AMA Style**

Postle B.
Methods for Creating Curved Shell Structures From Sheet Materials. *Buildings*. 2012; 2(4):424-455.
https://doi.org/10.3390/buildings2040424

**Chicago/Turabian Style**

Postle, Bruno.
2012. "Methods for Creating Curved Shell Structures From Sheet Materials" *Buildings* 2, no. 4: 424-455.
https://doi.org/10.3390/buildings2040424