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Ruled surface

 

 
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Ruled surface
 
 
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In geometryGeometry Summary

Geometry arose as the field of knowledge dealing with spatial relationships....
, a surfaceSurface

In mathematics, specifically in topology, a surface is a two-dimensional manifold....
  is ruled if through every point of there is a straight line that lies on . The most familiar examples are the planePlane (mathematics)

In mathematics, a plane is a fundamental two-dimensional object....
 and the curved surface of a cylinderCylinder (geometry)

In mathematics, a cylinder is a quadric, i.e....
 or cone. A ruled surface can be visualisedScientific visualization

Scientific-, and Information visualization are branches of computer graphics and user interface which are concerned wi...
 as the surface formed by moving a "straight" line in space. For example, a cone is formed by keeping one end-pointPoint (geometry) Overview

A spatial point is an entity with a location in space but no extent....
 of a line fixed whilst moving the other end-point in a circleCircle

In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed poi...
.

The properties of being ruled is preserved by projective maps, and therefore is a concept of projective geometryProjective geometry Summary

Projective geometry is a non-metrical form of geometry that emerged in the early 19th century....
. Analogues for algebraic surfaceAlgebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two....
s are studied in algebraic geometryAlgebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative...
.

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

...
 of the form

where is a curve lying in , and is a curve on the unit sphere. Thus, for example, if one uses

one obtains a ruled surface that contains the Möbius stripMöbius strip

The Mbius strip or Mbius band is a surface with only one side and only one boundary component....
.

Alternatively, a ruled surface can be parametrizedParametric model

In statistics, a parametric model is a parametrized family of probability distributions, one of which is presumed to describ...
 as , where and are two non-intersecting curves lying on . In particular, when and move with constant speed along two skew linesSkew lines

In geometry, skew lines are two lines that do not intersect but are not parallel....
, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet.

Doubly ruled surface

A surface is doubly ruled if through every one of its points there are two distinct lines that lie on .

Particulars

  • The plane, which is also the only n-ally ruled surface for n = 3.
  • The hyperbolic paraboloid
  • The hyperboloid of one sheet

Application

Doubly ruled surfaces are used in the study of skew linesFacts About Skew lines

In geometry, skew lines are two lines that do not intersect but are not parallel....
. Many hyperboloid structureHyperboloid structure

Hyperboloid structures in architecture were first applied by Russian engineer Vladimir Shukhov....
s have been built making use of only straight materials.

Developable surface

A developable surfaceDevelopable surface

A developable surface is a surface with a metric that can be flattened onto a plane without distortion....
 — one that can be (locally) unrolled onto a flat plane without tearing or stretching — if complete, is necessarily ruled, but the converse is not always true. Thus the cylinder and cone are developable, but the general hyperboloid of one sheet is not.

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