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Ruled surface

 
Ruled Surface

 
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Ruled surface
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, 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 E3....
  is ruled if through every point of there is a straight line that lies on . The most familiar examples are the plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
 and the curved surface of a cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
 or cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
. Other examples are a conical surface
Conical surface

In geometry, a conical surface is the unbounded surface formed by the union of all the straight line 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....
 with elliptical
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 directrix, the right conoid, the helicoid
Helicoid

The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776....
, and the tangent developable
Tangent developable

The tangent developable of a space curve is a ruled surface of the form . Intuitively it is the union of the tangent lines to the curve.A result of Euler states that most developable surfaces can be obtained as a tangent developable....
 of a smooth curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 in space.

A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, 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 E3....
  is ruled if through every point of there is a straight line that lies on . The most familiar examples are the plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
 and the curved surface of a cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
 or cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
. Other examples are a conical surface
Conical surface

In geometry, a conical surface is the unbounded surface formed by the union of all the straight line 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....
 with elliptical
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 directrix, the right conoid, the helicoid
Helicoid

The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776....
, and the tangent developable
Tangent developable

The tangent developable of a space curve is a ruled surface of the form . Intuitively it is the union of the tangent lines to the curve.A result of Euler states that most developable surfaces can be obtained as a tangent developable....
 of a smooth curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 in space.

A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line. For example, a cone is formed by keeping one end-point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 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 point in a plane which are the same distance from a given point called the center....
.

A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains three distinct lines through each of its points.

The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry
Projective geometry

In mathematics projective geometry is the study of geometric properties which are invariant under projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry and projective differential geometry ....
. Analogues for algebraic surface
Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension of an algebraic variety two. In the case of geometry over the field of complex number, an algebraic surface is therefore of complex dimension two and so of dimension four as a smooth manifold....
s are studied in algebraic geometry
Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry....
.

Parametric representation

The "moving line" view means that a ruled surface has a parametric representation
Parametric equation

In mathematics, parametric equations are a method of defining a curve. A simple kinematics example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
 of the form where is the generic point on the surface, is point that traces a curve lying on the surface, and is a unit-length vector
Unit vector

In mathematics, a unit vector in a normed vector space is a Vector space whose Norm is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or ?hat?, like this: ....
 that traces a curve on the unit sphere. Thus, for example, if one uses

one obtains a ruled surface that contains the Möbius strip
Möbius strip

The M?bius strip or M?bius band is a surface with only one side and only one boundary component. The M?bius strip has the mathematical property of being orientability....
.

Alternatively, a ruled surface can be parametrized
Parametric model

Definition. A set of probability measures on indexed by a parameter is said to be a parametric model or parametric family if and only if the parameter space is a subset of ....
 as , where and are two non-intersecting curves lying on the surface. In particular, when and move with constant speed along two skew lines
Skew lines

In solid geometry, skew lines are two lines that do not intersect but are not parallel. Equivalently, they are lines that are not both in the same plane ....
, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet.

Ruled Hyperboloid

Developable surface


A developable surface
Developable surface

In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is "surface" that can be Flatness onto a Plane without distortion ....
 is a surface that can be (locally) unrolled onto a flat plane without tearing or stretching it. If a developable surface lies in three-dimensional Euclidean space, and is complete
Complete space

In mathematical analysis, a metric space M is said to be complete if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M....
, then it is necessarily ruled, but the converse is not always true. For instance, the cylinder and cone are developable, but the general hyperboloid of one sheet is not. More generally, any developable surface in three-dimensions is part of a complete ruled surface, and so itself must be locally ruled. There are surfaces embedded in four dimensions which are not ruled, however .

Applications

Doubly-ruled surfaces are the inspiration for curved hyperboloid structure
Hyperboloid structure

Hyperboloid structures are architectural structures designed with hyperboloid geometry. Often these are tall structures such as towers where the hyperboloid geometry's structural strength is used to support an object high off the ground, but hyperboloid geometry is also often used for decorative effect as well as structural economy....
s that can be built with a lattice
Lattice (mathematics)

In mathematics, the term lattice can mean:* A partially ordered set in which any two elements have a supremum and an infimum—see lattice ....
 of straight elements, namely:
  • Hyperbolic paraboloids, such as saddle roof
    Saddle roof

    A saddle roof is one which follows a convex curve about one axis and a concave curve about the other. The hyperbolic paraboloid form has been used for roofs at various times since it is easily constructed from straight sections of lumber, steel, or other conventional materials....
    s.
  • Hyperboloids of one sheet, such as cooling tower
    Cooling tower

    Cooling towers are heat removal devices used to transfer process waste heat to the atmosphere. Cooling towers may either use the evaporation of water to remove process heat and cool the working fluid to near the Wet-bulb temperature or rely solely on air to cool the working fluid to near the Dry-bulb temperature....
    s and some trash bin
    Waste container

    A waste container is a container for temporarily storing waste, which is usually made out of metal or plastic. Common terms are dustbin, 'rubbish bin, 'litter bin, 'garbage can, 'trash can, 'trash bin, 'dumpster, 'Container Bin, 'Bin 'trash barrel, and rubbish barrel; the word can...
    s.


The RM-81 Agena
RM-81 Agena

The Agena was a rocket upper stage developed by Lockheed Corporation for the ill-fated WS-117L US reconnaissance satellite program. It lived on to see extensive use as the upper stage/spacecraft for the Corona spy satellite program and as an upper stage on the Thor , Atlas , and Titan boosters....
 rocket engine
Rocket engine

A rocket engine or simply rocket is a jet engineRocket Propulsion Elements; 7th edition- chapter 1 that uses only propellant mass for forming its high speed propulsive Jet ....
 employed straight cooling channels that were laid out in a ruled surface to form the throat of the nozzle
Nozzle

A nozzle is a mechanical device designed to control the characteristics of a fluid flow as it exits an enclosed chamber or pipe via an orifice....
 section.

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