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Developable surface

 

Developable surface

In mathematics, a developable surface (or torse: archaic) is a surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. "stretching" or "compressing"). 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 R4 which are not ruled.[1]

Contents

  • Particulars 1
    • Application 1.1
  • Non-developable surface 2
    • Applications of non-developable surfaces 2.1
  • See also 3
  • References 4
  • External links 5

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.
  • 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.

Application

Developable surfaces have several practical applications. Many cartographic projections involve projecting the Earth to a developable surface and then "unrolling" the surface into a region on the plane. Since they 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.[2]

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.

See also

References

  1. ^  
  2. ^ Nolan, T. J. (1970), Computer-Aided Design of Developable Hull Surfaces, Ann Arbor: University Microfilms International 

External links

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