# Developable surface

In mathematics, a **developable surface** (or **torse**: archaic) is a surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. "stretching" or "compressing"). Conversely, it is a surface which can be made by transforming a plane (i.e. "folding", "bending", "rolling", "cutting" and/or "gluing"). In three dimensions all developable surfaces are ruled surfaces (but not vice versa). There are developable surfaces in **R**^{4} which are not ruled.^{[1]}

## Particulars

The developable surfaces which can be realized in three-dimensional space include:

- Cylinders and, more generally, the "generalized" cylinder; its cross-section may be any smooth curve
- Cones and, more generally, conical surfaces; away from the apex
- The oloid and the sphericon are members of a special family of solids that develop their entire surface when rolling down a flat plane.
- Planes (trivially); which may be viewed as a cylinder whose cross-section is a line
- Tangent developable surfaces; which are constructed by extending the tangent lines of a spatial curve.
- The torus has a metric under which it is developable, which can be embedded into three-dimensional space by the Nash embedding theorem
^{[2]}and has a simple representation in four dimensions as the Cartesian product of two circles: see Clifford torus.

Formally, in mathematics, a developable surface is a surface with zero Gaussian curvature. One consequence of this is that all "developable" surfaces embedded in 3D-space are ruled surfaces (though hyperboloids are examples of ruled surfaces which are not developable). Because of this, many developable surfaces can be visualised as the surface formed by moving a straight line in space. For example, a cone is formed by keeping one end-point of a line fixed whilst moving the other end-point in a circle.

### Application

Developable surfaces have several practical applications. Many cartographic projections involve projecting the Earth to a developable surface and then "unrolling" the surface into a region on the plane. Since they may be constructed by bending a flat sheet, they are also important in manufacturing objects from sheet metal, cardboard, and plywood. An industry which uses developed surfaces extensively is shipbuilding.^{[3]}

## Non-developable surface

Most smooth surfaces (and most surfaces in general) are not developable surfaces. **Non-developable surfaces** are variously referred to as having "**double curvature**", "**doubly curved**", "**compound curvature**", "**non-zero Gaussian curvature**", etc.

Some of the most often-used non-developable surfaces are:

- Spheres are not developable surfaces under any metric as they cannot be unrolled onto a plane.
- The helicoid is a ruled surface – but unlike the ruled surfaces mentioned above, it is not a developable surface.
- The hyperbolic paraboloid and the hyperboloid are slightly different doubly ruled surfaces – but unlike the ruled surfaces mentioned above, neither one is a developable surface.

### Applications of non-developable surfaces

Many gridshells and tensile structures and similar constructions gain strength by using (any) doubly curved form.

## See also

## References

- ↑ Hilbert, David; Cohn-Vossen, Stephan (1952),
*Geometry and the Imagination*(2nd ed.), New York: Chelsea, pp. 341–342, ISBN 978-0-8284-1087-8 - ↑ Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration",
*Proceedings of the National Academy of Sciences*, Proceedings of the National Academy of Sciences,**109**(19): 7218–7223, doi:10.1073/pnas.1118478109. - ↑ Nolan, T. J. (1970),
*Computer-Aided Design of Developable Hull Surfaces*, Ann Arbor: University Microfilms International

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