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 surface. There are developable surface in R⁴ which are not ruled.^[1]

Particulars

The "developable" surfaces which can be realized in 3D-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).

See also

• Development (differential geometry)

References

1. ^ Hilbert, David & Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, pp. 341-342, ISBN 978-0-8284-1087-8

• Eric W. Weisstein, Developable Surface at MathWorld.
• Examples of developable surfaces on the Rhino3DE website