mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a developable surface (or torse: archaic) is a smooth

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

with zero

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

. That is, it is a surface that can be flattened onto a plane without

distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...

(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"). Because of these properties, developable surfaces are widely used in the design and fabrication of items to be made from sheet materials, ranging from

textiles Textile is an Hyponymy and hypernymy, umbrella term that includes various Fiber, fiber-based materials, including fibers, yarns, Staple (textiles)#Filament fiber, filaments, Thread (yarn), threads, and different types of #Fabric, fabric. ...

sheet metal Sheet metal is metal formed into thin, flat pieces, usually by an industrial process. Thicknesses can vary significantly; extremely thin sheets are considered foil (metal), foil or Metal leaf, leaf, and pieces thicker than 6 mm (0.25 ...

such as ductwork to

shipbuilding Shipbuilding is the construction of ships and other Watercraft, floating vessels. In modern times, it normally takes place in a specialized facility known as a shipyard. Shipbuilders, also called shipwrights, follow a specialized occupation th ...

. In

three dimensions In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position of a point. Most commonly, it is the three-di ...

all developable surfaces are

ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...

s (but not vice versa). There are developable surfaces in

four-dimensional space Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...

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 (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...

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

Particulars

The developable surfaces which can be realized in

three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...

include: *

Cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...

s and, more generally, the "generalized" cylinder; its

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

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 ...

Cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

s and, more generally, conical surfaces; away from the

apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics) A-Bomb Abomination Absorbing Man Abraxas Abyss Abyss is the name of two characters appearing in Ameri ...

* The oloid and the

sphericon In solid geometry, the sphericon is a solid that has a continuous developable surface with two Congruence (geometry), congruent, semicircle, semi-circular edges, and four Vertex (geometry), vertices that define a square. It is a member of a spe ...

are members of a special family of solids that develop their entire surface when

rolling Rolling is a Motion (physics)#Types of motion, type of motion that combines rotation (commonly, of an Axial symmetry, axially symmetric object) and Translation (geometry), translation of that object with respect to a surface (either one or the ot ...

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

lines of a spatial curve. * The

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

has a metric under which it is developable, which can be embedded into three-dimensional space by the

Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedding, embedded into some Euclidean space. Isometry, Isometric means preserving the length of ever ...

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 and (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Cliffo ...

. 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 def ...

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 point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

Application

Developable surfaces have several practical applications. Many cartographic projections involve projecting the

Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

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 the secondary sector of the economy. The term may refer ...

objects from sheet metal,

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

, and

plywood Plywood is a composite material manufactured from thin layers, or "plies", of wood veneer that have been stacked and glued together. It is an engineered wood from the family of manufactured boards, which include plywood, medium-density fibreboa ...

. An

industry Industry may refer to: Economics * Industry (economics), a generally categorized branch of economic activity * Industry (manufacturing), a specific branch of economic activity, typically in factories with machinery * The wider industrial sector ...

which uses developed surfaces extensively is shipbuilding. Developable Mechanisms are mechanisms that conform to a developable surface and can exhibit motion (deploy) off the surface.

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 (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, 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 Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

as they cannot be unrolled onto a plane. * The

helicoid The helicoid, also known as helical surface, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation ...

is a ruled surface – but unlike the ruled surfaces mentioned above, it is not a developable surface. * The

hyperbolic paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every pla ...

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