Developable surface



In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). 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 four-dimensional space $\mathbb{R}^4$ which are not ruled.

The envelope 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:


 * 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 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.

Developable Mechanisms are mechanisms that conform to a developable surface and can exhibit motion (deploy) 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 developable surfaces 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 (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.