Developable Surface - Particulars

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 is one of very few geometrical objects that develops its 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.

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 (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.