World Library  
Flag as Inappropriate
Email this Article

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]


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


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.


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


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

External links

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.