# Developable surface

In mathematics, a **developable surface** is a surface with zero Gaussian curvature. That is, it is "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. There are developable surfaces in **R**^{4} which are not ruled.^{[1]}

## Particulars

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

- 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
- 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 spacial curve.

Spheres are not developable surfaces under any metric as they cannot be unrolled onto a plane. The torus has a metric under which it is developable, but such a torus does not embed into 3D-space. It can, however, be realized in four dimensions.

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^{[2]}).

## See also

- Development (differential geometry)

## 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 - ↑ Nolan, T.J. (1970),
*Computer-Aided Design of Developable Hull Surfaces*, Ann Arbor: University Microfilms International