Ruled surface - Revision history

Vipul: /* Examples */

2011-08-05T14:50:16Z

Examples

← Older revision		Revision as of 14:50, 5 August 2011
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	\| [[elliptic hyperboloid of one sheet]] \|\| <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1</math> \|\| \|\|		\| [[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> \|\| \|\|		\| [[hyperbolic paraboloid]] \|\| <math>z = \frac{y^2}{b^2} - \frac{x^2}{a^2}</math> \|\| \|\| This is in fact a [[doubly ruled surface]]
	\|-		\|-
	\| [[helicoid]] \|\| <math>y = x \tan(z/c)</math> \|\| \|\| it is the only ruled [[minimal surface]] other than the plane.		\| [[helicoid]] \|\| <math>y = x \tan(z/c)</math> \|\| \|\| it is the only ruled [[minimal surface]] other than the plane.

Vipul: /* Examples */

2011-08-05T14:49:04Z

Examples

← Older revision		Revision as of 14:49, 5 August 2011
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	! Ruled surface !! Equational/implicit description !! Functions <math>\mathbf{b}</math> and <math>\delta</math> in a possible parametric description !! Comment		! 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]].		\| [[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]] and also a [[minimal 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>. \|\|		\| [[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>. \|\|
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	\| [[helicoid]] \|\| <math>y = x \tan(z/c)</math> \|\| \|\| it is the only ruled [[minimal surface]] other than the plane.		\| [[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==

Vipul at 14:48, 5 August 2011

2011-08-05T14:48:41Z

@@ Line 3: / Line 3: @@
 ==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.
 ==Relation with other properties==
 ===Stronger properties===
-* [[Doubly ruled surface]]
+* [[Weaker than::Doubly ruled surface]]
-* [[Developable surface]]
+* [[Weaker than::Developable surface]]

Vipul: 1 revision

2008-05-18T20:07:50Z

1 revision

← Older revision	Revision as of 20:07, 18 May 2008
(No difference)

Vipul at 15:38, 6 July 2007

2007-07-06T15:38:40Z

New page

{{surface-in-3D property}}

==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.

==Relation with other properties==

===Stronger properties===

* [[Doubly ruled surface]]
* [[Developable surface]]