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. 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. Developable surfaces have several practical applications.