# Difference between revisions of "Ruled surface"

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==Definition== | ==Definition== | ||

− | A '''ruled surface''' is a surface in <math>\R^3</math> with the property that for any point on the surface, there is a line through that point lying on the surface. | + | A '''ruled surface''' is a [[surface]] in [[Euclidean space]] <math>\R^3</math> with the property that for any point on the surface, there is a line through that point lying on the surface. |

+ | Because, by definition, a surface is connected, an alternative and equivalent definition of ruled surface is as a surface that can be ''swept'' by moving a line in space. | ||

+ | |||

+ | ===Parametric description=== | ||

+ | |||

+ | A ruled surface can be described by a parametric description of the form: | ||

+ | |||

+ | <math>\! \mathbf{x}(u,v) = \mathbf{b}(u) + v \delta(u)</math> | ||

+ | |||

+ | where <math>\mathbf{b}</math> and <math>\delta</math> are functions that take scalar inputs and give three-dimensional vector outputs. Here, the parameter <math>u</math> controls which line we are on, and the parameter <math>v</math> describes the location of the point on the line. In other words, for every fixed value of <math>u</math>, we get a fixed line described with a single parameter <math>v</math>. The surface is the union of these lines. | ||

+ | |||

+ | We use the following terminology: | ||

+ | |||

+ | * The line for each fixed value of <math>u</math> is termed a ''ruling'' for the surface. | ||

+ | * The function <math>\mathbf{b}</math> is termed the '''ruled surface directrix''' or the base curve. For any <math>u</math>, <math>\mathbf{b}(u)</math> describes the ''position'' of the line. | ||

+ | * The value <math>\delta(u)</math> describes a ''direction'' vector along the line, and the function <math>\delta</math> is termed a '''director curve'''. | ||

+ | |||

+ | ==Examples== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Ruled surface !! Equational/implicit description !! Functions <math>\mathbf{b}</math> and <math>\delta</math> in a possible parametric description !! Comment | ||

+ | |- | ||

+ | | [[Euclidean plane]] || <math>z = 0</math> (the <math>xy</math>-plane) || <math>\mathbf{b}(u)</math> is the vector with coordinates <math>(u,0,0)</math> and <math>\delta(u)</math> is the vector <math>(0,1,0)</math>. || The Euclidean plane is in fact a [[doubly ruled surface]]. | ||

+ | |- | ||

+ | | [[right circular cylinder]] (infinite version) || <math>x^2 + y^2 = 1</math> (the right circular cylinder with base circle the unit circle in the <math>xy</math>-plane and axis along the <math>z</math>-axis). || <math>\mathbf{b}(u)</math> is the vector with coordinates <math>(\cos u, \sin u,0)</math> and <math>\delta(u)</math> is the vector <math>(0,0,1)</math>. || | ||

+ | |- | ||

+ | | [[circular hyperboloid of one sheet]] || <math>\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1</math> || || This is in fact a [[doubly ruled surface]] | ||

+ | |- | ||

+ | | [[elliptic hyperboloid of one sheet]] || <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1</math> || || | ||

+ | |- | ||

+ | | [[hyperbolic paraboloid]] || <math>z = \frac{y^2}{b^2} - \frac{x^2}{a^2}</math> || || | ||

+ | |- | ||

+ | | [[helicoid]] || <math>y = x \tan(z/c)</math> || || it is the only ruled [[minimal surface]] other than the plane. | ||

+ | |} | ||

==Relation with other properties== | ==Relation with other properties== | ||

===Stronger properties=== | ===Stronger properties=== | ||

− | * [[Doubly ruled surface]] | + | * [[Weaker than::Doubly ruled surface]] |

− | * [[Developable surface]] | + | * [[Weaker than::Developable surface]] |

## Revision as of 14:48, 5 August 2011

This article defines a property that makes sense for a surface embedded in , viz three-dimensional Euclidean space. The property is invariant under orthogonal transformations and scaling, i.e., under all similarity transformations.

View other such properties

## Contents

## Definition

A **ruled surface** is a surface in Euclidean space with the property that for any point on the surface, there is a line through that point lying on the surface.

Because, by definition, a surface is connected, an alternative and equivalent definition of ruled surface is as a surface that can be *swept* by moving a line in space.

### Parametric description

A ruled surface can be described by a parametric description of the form:

where and are functions that take scalar inputs and give three-dimensional vector outputs. Here, the parameter controls which line we are on, and the parameter describes the location of the point on the line. In other words, for every fixed value of , we get a fixed line described with a single parameter . The surface is the union of these lines.

We use the following terminology:

- The line for each fixed value of is termed a
*ruling*for the surface. - The function is termed the
**ruled surface directrix**or the base curve. For any , describes the*position*of the line. - The value describes a
*direction*vector along the line, and the function is termed a**director curve**.

## Examples

Ruled surface | Equational/implicit description | Functions and in a possible parametric description | Comment |
---|---|---|---|

Euclidean plane | (the -plane) | is the vector with coordinates and is the vector . | The Euclidean plane is in fact a doubly ruled surface. |

right circular cylinder (infinite version) | (the right circular cylinder with base circle the unit circle in the -plane and axis along the -axis). | is the vector with coordinates and is the vector . | |

circular hyperboloid of one sheet | This is in fact a doubly ruled surface | ||

elliptic hyperboloid of one sheet | |||

hyperbolic paraboloid | |||

helicoid | it is the only ruled minimal surface other than the plane. |