A developable surface is a surface that can be flattened onto a plane without distortion (i.e. stretching, compressing, tearing). Inversely, it is a surface that can be made by transforming a plane (i.e. folding, bending, rolling). An open surface with X-, Y-, and Z-contours shown. ...

The developable surfaces in 3D are:

• cylinders and, more generally, the generalized cylinder: the cross-section can be any smooth curve
• cones and, more generally, conical surfaces, away from the apex
• (trivially:) planes

Spheres are not developable surfaces as they cannot be unrolled into a plane. A right circular cylinder In mathematics, a cylinder is a quadric, i. ... In common usage and elementary geometry, a cone (Greek: κώνος) is a solid object obtained by rotating a right triangle around one of its two short sides, the cones axis. ... 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 — that does not contain the apex. ... A sphere is a perfectly symmetrical geometrical object. ...

Formally, in mathematics, a developable surface is a surface with zero Gaussian curvature; that is, for every point on the surface, there is a straight line on the surface that passes through that point. This is also known as being linear in one direction. A plane is linear in all directions; a cylinder is linear in one and curved in the other; a sphere is curved in two directions. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Curvature is the amount by which a geometric object deviates from being flat. ...

Because a developable surface is linear in one direction, it can be visualised as the surfaced formed by moving a straight line in space. For example, a cone is formed by keeping one end of a line fixed while moving the other end in a circle. More complex developable surfaces may be formed by an intersection of sub-surfaces (formed by moving different lines).

Developable surfaces are important for various applications. They allow a mapping to a plane which locally preserves angles and distances. In cartography, for mapping part of the Earth to a plane (map projections) they can be used as intermediate stage. They are also important in manufacturing objects from sheet metal, cardboard, etc. Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ... A map projection is any of many methods used in cartography (mapmaking) to represent the two-dimensional curved surface of the earth or other body on a plane. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Sheet metal is simply metal formed into thin and flat pieces. ... Cardboard (called corrugated paper in the industry) is a heavy wood-based type of paper, notable for its stiffness and durability. ...

See also

• Ruled surface

In geometry, a surface is ruled if through every point of there is a straight line that lies on . ...

External Links

• Examples of developable surfaces on the Rhino3DE website

References

• Developable Surface entry at MathWorld