Developable surface

In mathematics, a developable surface 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. There are developable surfaces in R^{4} which are not ruled.^{[1]}
Contents
Particulars
The developable surfaces which can be realized in threedimensional space are^{[2]}:
 Cylinders and, more generally, the "generalized" cylinder; its crosssection may be any smooth curve
 Cones and, more generally, conical surfaces; away from the apex
 Planes (trivially); which may be viewed as a cylinder whose crosssection is a line
 Tangent developable surfaces; which are constructed by extending the tangent lines of a spatial curve.
Spheres are not developable surfaces under any metric as they cannot be unrolled onto a plane. The torus has a metric under which it is developable, but such a torus does not embed into 3Dspace. 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 3Dspace 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 endpoint of a line fixed whilst moving the other endpoint 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^{[3]}).
See also
References
 ^ Hilbert, David; CohnVossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, pp. 341–342, ISBN 9780828410878
 ^ Stoker, J.J. (1961), Developable surfaces in the large. Comm. Pure Appl. Math. 14(3), 627635, doi:10.1002/cpa.3160140333
 ^ Nolan, T.J. (1970), ComputerAided Design of Developable Hull Surfaces, Ann Arbor: University Microfilms International
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Developable surface — Developable De*vel op*a*ble, a. Capable of being developed. J. Peile. [1913 Webster] {Developable surface} (Math.), a surface described by a moving right line, and such that consecutive positions of the generator intersect each other. Hence, the… … The Collaborative International Dictionary of English
developable surface — noun A surface (in more than two dimensions, a plane is trivially developable) that can be flattened into a plane without distortion … Wiktionary
developable surface — noun Etymology: translation of French surface développable : a surface that may be imagined flattened out upon a plane without stretching any element * * * Math. a surface that can be flattened onto a plane without stretching or compressing any… … Useful english dictionary
developable surface — Math. a surface that can be flattened onto a plane without stretching or compressing any part of it, as a circular cone. * * * … Universalium
Developable — De*vel op*a*ble, a. Capable of being developed. J. Peile. [1913 Webster] {Developable surface} (Math.), a surface described by a moving right line, and such that consecutive positions of the generator intersect each other. Hence, the surface can… … The Collaborative International Dictionary of English
Developable — In mathematics, the term developable may refer to: A developable space in general topology. A developable surface in geometry. This disambiguation page lists articles associated with the same title. If an internal link … Wikipedia
developable — adj. that can be developed. Phrases and idioms: developable surface Geom. a surface that can be flattened into a plane without overlap or separation, e.g. a cylinder … Useful english dictionary
Ruled surface — A hyperboloid of one sheet is a doubly ruled surface: it can be generated by either of two families of straight lines. In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar… … Wikipedia
Tangential developable — In the mathematical study of the differential geometry of surfaces, a tangential developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a… … Wikipedia
Conical surface — A circular conical surface In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point the apex or vertex and any point of some fixed space curve the directrix… … Wikipedia