# Difference between revisions of "Doubly ruled surface"

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A surface embedded in <math>\R^3</math> is said to be '''doubly ruled''' if given any point in the surface, there are two distinct lines through the point, that lie completely on the surface. | A surface embedded in <math>\R^3</math> is said to be '''doubly ruled''' if given any point in the surface, there are two distinct lines through the point, that lie completely on the surface. | ||

+ | |||

+ | ==Classification== | ||

+ | |||

+ | {{further|[[Classification of doubly ruled surfaces]]}} | ||

+ | |||

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

+ | ! Doubly ruled surface !! Equational/implicit description | ||

+ | |- | ||

+ | | [[Euclidean plane]] || <math>z = 0</math> (the <math>xy</math>-plane) | ||

+ | |- | ||

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

+ | |- | ||

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

+ | |} | ||

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

## Latest revision as of 14:52, 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

## Definition

A surface embedded in is said to be **doubly ruled** if given any point in the surface, there are two distinct lines through the point, that lie completely on the surface.

## Classification

`Further information: Classification of doubly ruled surfaces`

Doubly ruled surface | Equational/implicit description |
---|---|

Euclidean plane | (the -plane) |

circular hyperboloid of one sheet | |

hyperbolic paraboloid |