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.
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==Classification==
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{{further|[[Classification of doubly ruled surfaces]]}}
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{| class="sortable" border="1"
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! Doubly ruled surface !! Equational/implicit description
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|-
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| [[Euclidean plane]] || <math>z = 0</math> (the <math>xy</math>-plane)
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|-
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| [[circular hyperboloid of one sheet]] || <math>\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1</math>
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|-
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| [[hyperbolic paraboloid]] || <math>z = \frac{y^2}{b^2} - \frac{x^2}{a^2}</math>
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|}
  
 
==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 \R^3, 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 \R^3 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 z = 0 (the xy-plane)
circular hyperboloid of one sheet \frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1
hyperbolic paraboloid z = \frac{y^2}{b^2} - \frac{x^2}{a^2}

Relation with other properties

Weaker properties