Definitions

developable-surface

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 surfaces in R4 which are not ruled.

Particulars

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

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

References

Search another word or see developable-surfaceon Dictionary | Thesaurus |Spanish
Copyright © 2014 Dictionary.com, LLC. All rights reserved.
  • Please Login or Sign Up to use the Recent Searches feature