The cylinder is an example of a 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 which are not ruled.[1]

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:

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.


Comparison of tangent and secant cylindrical, conic and azimuthal map projections with standard parallels shown in red

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 cartographic projections involve projecting the Earth to a developable surface and then "unrolling" the surface into a region on the plane.

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:

Applications of non-developable surfaces

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

See also


  1. ^ Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, pp. 341–342, ISBN 978-0-8284-1087-8
  2. ^ Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC 3358891, PMID 22523238.
  3. ^ "Developable Mechanisms | About Developable Mechanisms". compliantmechanisms. Retrieved 2019-02-14.
  4. ^ Howell, Larry L.; Lang, Robert J.; Magleby, Spencer P.; Zimmerman, Trent K.; Nelson, Todd G. (2019-02-13). "Developable mechanisms on developable surfaces". Science Robotics. 4 (27): eaau5171. doi:10.1126/scirobotics.aau5171. ISSN 2470-9476. PMID 33137737.
  5. ^ Nolan, T. J. (1970), Computer-Aided Design of Developable Hull Surfaces, Ann Arbor: University Microfilms International