Doubly ruled surface: Difference between revisions

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 $x y$ -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

Ruled surface

@@ Line 4: / Line 4: @@
 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==