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.
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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.
  
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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.
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===Parametric description===
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A ruled surface can be described by a parametric description of the form:
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<math>\! \mathbf{x}(u,v) = \mathbf{b}(u) + v \delta(u)</math>
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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.
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We use the following terminology:
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* The line for each fixed value of <math>u</math> is termed a ''ruling'' for the surface.
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* 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.
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* The value <math>\delta(u)</math> describes a ''direction'' vector along the line, and the function <math>\delta</math> is termed a '''director curve'''.
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==Examples==
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{| class="sortable" border="1"
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! Ruled surface !! Equational/implicit description !! Functions <math>\mathbf{b}</math> and <math>\delta</math> in a possible parametric description !! Comment
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|-
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| [[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]].
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|-
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| [[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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|-
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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> || || This is in fact a [[doubly ruled surface]]
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|-
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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> || ||
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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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|-
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| [[helicoid]] || <math>y = x \tan(z/c)</math> || || it is the only ruled [[minimal surface]] other than the plane.
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|}
 
==Relation with other properties==
 
==Relation with other properties==
  
 
===Stronger properties===
 
===Stronger properties===
  
* [[Doubly ruled surface]]
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* [[Weaker than::Doubly ruled surface]]
* [[Developable surface]]
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* [[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 \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 ruled surface is a surface in Euclidean space \R^3 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:

\! \mathbf{x}(u,v) = \mathbf{b}(u) + v \delta(u)

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

We use the following terminology:

  • The line for each fixed value of u is termed a ruling for the surface.
  • The function \mathbf{b} is termed the ruled surface directrix or the base curve. For any u, \mathbf{b}(u) describes the position of the line.
  • The value \delta(u) describes a direction vector along the line, and the function \delta is termed a director curve.

Examples

Ruled surface Equational/implicit description Functions \mathbf{b} and \delta in a possible parametric description Comment
Euclidean plane z = 0 (the xy-plane) \mathbf{b}(u) is the vector with coordinates (u,0,0) and \delta(u) is the vector (0,1,0). The Euclidean plane is in fact a doubly ruled surface.
right circular cylinder (infinite version) x^2 + y^2 = 1 (the right circular cylinder with base circle the unit circle in the xy-plane and axis along the z-axis). \mathbf{b}(u) is the vector with coordinates (\cos u, \sin u,0) and \delta(u) is the vector (0,0,1).
circular hyperboloid of one sheet \frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1 This is in fact a doubly ruled surface
elliptic hyperboloid of one sheet \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1
hyperbolic paraboloid z = \frac{y^2}{b^2} - \frac{x^2}{a^2}
helicoid y = x \tan(z/c) it is the only ruled minimal surface other than the plane.

Relation with other properties

Stronger properties