In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a developable surface is a surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 with zero Gaussian curvature

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

. That is, it is a "surface" that can be flattened

Flatness

Flatness may refer to:*Flatness *Flatness *Flatness *Flatness *Flatness , a geometrical tolerance required in certain manufacturing situations*Flatness...

 onto a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 without distortion

Distortion

A distortion is the alteration of the original shape of an object, image, sound, waveform or other form of information or representation. Distortion is usually unwanted, and often many methods are employed to minimize it in practice...

 (i.e. "stretching" or "compressing"). Conversely, it is a surface which can be made by transforming

Transformation (mathematics)

In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...

 a plane (i.e. "folding", "bending", "rolling", "cutting" and/or "gluing"). In three dimensions all developable surfaces are ruled surface

Ruled surface

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 examples are the plane and the curved surface of a cylinder or cone...

s. There are developable surfaces in R⁴ which are not ruled.

Particulars

The developable surfaces which can be realized in three-dimensional space

Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

 are:

• Cylinder
  Cylinder (geometry)
  A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
  
  s and, more generally, the "generalized" cylinder; its cross-section
  Cross section (geometry)
  In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
  
   may be any smooth
  Smooth function
  In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
  
   curve
  Curve
  In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
  
  
• Cone
  Cone (geometry)
  A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
  
  s and, more generally, conical surface
  Conical surface
  In geometry, a 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...
  
  s; away from the apex
  Apex (geometry)
  In geometry, an apex is the vertex which is in some sense the highest of the figure to which it belongs.*In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side....
  
  
• Planes (trivially); which may be viewed as a cylinder whose cross-section is a line
  Line (mathematics)
  The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
  
  
• Tangent developable
  Tangent developable
  The tangent developable of a space curve \gamma is a ruled surface of the form \gamma+s \gamma^\prime. Intuitively it is the union of the tangent lines to the space curve....
  
   surfaces; which are constructed by extending the tangent
  Tangent
  In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
  
   lines of a spatial curve.

Sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

s are not developable surfaces under any metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

 as they cannot be unrolled onto a plane. The torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

 has a metric under which it is developable, but such a torus does not embed

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

 into 3D-space. It can, however, be realized in four dimensions (see: Clifford torus

Clifford torus

In geometric topology, the Clifford torus is a special kind of torus sitting inside R4. Alternatively, it can be seen as a torus sitting inside C2 since C2 is topologically the same space as R4...

).

Formally, in mathematics, a developable surface is a surface with zero Gaussian curvature

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

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

Ruled surface

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 examples are the plane and the curved surface of a cylinder or cone...

s (though hyperboloids are examples of ruled surfaces which are not developable). Because of this, many developable surfaces can be visualised

Scientific visualization

Scientific visualization is an interdisciplinary branch of science according to Friendly "primarily concerned with the visualization of three-dimensional phenomena , where the emphasis is on realistic renderings of volumes, surfaces, illumination sources, and so forth, perhaps...

 as the surface formed by moving a straight line in space. For example, a cone is formed by keeping one end-point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

 of a line fixed whilst moving the other end-point in a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

Application

Developable surfaces have several practical applications. Many cartographic

Cartography

Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

 projections involve projecting the Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

 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

Manufacturing

Manufacturing is the use of machines, tools and labor to produce goods for use or sale. The term may refer to a range of human activity, from handicraft to high tech, but is most commonly applied to industrial production, in which raw materials are transformed into finished goods on a large scale...

 objects from sheet metal

Sheet metal

Sheet metal is simply metal formed into thin and flat pieces. It is one of the fundamental forms used in metalworking, and can be cut and bent into a variety of different shapes. Countless everyday objects are constructed of the material...

, cardboard, and plywood

Plywood

Plywood is a type of manufactured timber made from thin sheets of wood veneer. It is one of the most widely used wood products. It is flexible, inexpensive, workable, re-usable, and can usually be locally manufactured...

 (an industry

Industry

Industry refers to the production of an economic good or service within an economy.-Industrial sectors:There are four key industrial economic sectors: the primary sector, largely raw material extraction industries such as mining and farming; the secondary sector, involving refining, construction,...

 which uses developed surfaces extensively is shipbuilding

Shipbuilding

Shipbuilding is the construction of ships and 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 roots to before recorded history.Shipbuilding and ship repairs, both...

).