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. "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.
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 and the sphericon are members of a special family of 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 torus has a metric under which it is developable, which can be embedded into three-dimensional space by the Nash embedding theorem 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.
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.
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
- Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, pp. 341–342, ISBN 978-0-8284-1087-8
- 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, Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC , PMID 22523238.
- Nolan, T. J. (1970), Computer-Aided Design of Developable Hull Surfaces, Ann Arbor: University Microfilms International
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