Developable surface

In

${\displaystyle \mathbb {R} ^{4}}$ 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:

• smooth curve
• Cones and, more generally, conical surfaces; away from the apex
• The
  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
  Nash embedding theorem^[2] 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.^[3][4]

Many

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.^[5]

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.

Many gridshells and tensile structures and similar constructions gain strength by using (any) doubly curved form.

See also

• Development (differential geometry)
• Developable roller

References

External links

Wikimedia Commons has media related to Developable surfaces.

• Weisstein, Eric W. "Developable Surface". MathWorld.
• Examples of developable surfaces on the Rhino3DE website