Developable surface

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 R4 which are not ruled.[1]

Contents

Particulars

The developable surfaces which can be realized in three-dimensional space are[2]:

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 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[3]).

See also

References

  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. ^ Stoker, J.J. (1961), Developable surfaces in the large. Comm. Pure Appl. Math. 14(3), 627--635, doi:10.1002/cpa.3160140333
  3. ^ Nolan, T.J. (1970), Computer-Aided 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

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