Developable surface
From Wikipedia, the free encyclopedia
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, compressing, tearing). Inversely, it is a surface that can be made by transforming a plane (i.e. folding, bending, rolling, cutting, and gluing).
The developable surfaces that can be realized in 3D space are:
- cylinders and, more generally, the generalized cylinder: the cross-section can be any smooth curve
- cones and, more generally, conical surfaces, away from the apex
- (trivially:) planes, which can be viewed as a cylinder whose cross-section is a line
Spheres are not developable surfaces under any metric as they cannot be unrolled into a plane. The torus has a metric under which it is developable, but such a torus does not embed into 3D space. It can 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 that 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 of a line fixed while moving the other end in a circle.
Developable surfaces have several applications. Many cartographic projections involve projecting the Earth to a developable surface and then unrolling the surface into a region in the plane. Since they can 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).
