In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

. 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 surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directri ...

s (but not vice versa). There are developable surfaces in four-dimensional space which are not ruled. The

envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...

of a single parameter family of planes is called a developable surface.

Particulars

The developable surfaces which can be realized in three-dimensional space include: * Cylinders and, more generally, the "generalized" cylinder; its

cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Ab ...

may be any smooth

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

* 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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

* Tangent developable surfaces; which are constructed by extending the

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

lines of a spatial curve. * The

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

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 In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon ...

. Formally, in mathematics, a developable surface is a surface with zero

. One consequence of this is that all "developable" surfaces embedded in 3D-space are

s (though

hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by de ...

s 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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

Application

Developable surfaces have several practical applications. Developable Mechanisms are mechanisms that conform to a developable surface and can exhibit motion (deploy) off the surface. Many cartographic projections involve projecting the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surf ...

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 Manufacturing is the creation or production of goods with the help of equipment, labor, machines, tools, and chemical or biological processing or formulation. It is the essence of secondary sector of the economy. The term may refer to a ...

objects from sheet metal,

cardboard Cardboard is a generic term for heavy paper-based products. The construction can range from a thick paper known as paperboard to corrugated fiberboard which is made of multiple plies of material. Natural cardboards can range from grey to light b ...

, and plywood. An industry which uses developed surfaces extensively is

shipbuilding Shipbuilding is the construction of ships and other Watercraft, floating vessels. It normally takes place in a specialized facility known as a shipyard. Shipbuilders, also called shipwrights, follow a specialized occupation that traces its roo ...

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: *

Sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

s 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

are slightly different doubly ruled surfaces – but unlike the ruled surfaces mentioned above, neither one is a developable surface.

Applications of non-developable surfaces

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

References

External links