Developable surface


In mathematics, a developable surface is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion. Conversely, it is a surface which can be made by transforming a plane. In three dimensions all developable surfaces are ruled surfaces. There are developable surfaces in R4 which are not ruled.
The envelop of a single parameter family of planes is called a developable surface.

Particulars

The developable surfaces which can be realized in three-dimensional space include:
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. 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.
Developable Mechanisms are mechanisms that conform to a developable surface and can exhibit motion off the surface.
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.

Non-developable surface

Most smooth surfaces 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:
Many gridshells and tensile structures and similar constructions gain strength by using doubly curved form.