Difference between revisions of "Ruled surface"
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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 | ||
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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]]. | + | | [[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]]. |
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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>. || | | [[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. | ||
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==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 14:49, 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
Contents
Definition
A ruled surface is a surface in Euclidean space 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:
where 
and 
are functions that take scalar inputs and give three-dimensional vector outputs. Here, the parameter 
controls which line we are on, and the parameter 
describes the location of the point on the line. In other words, for every fixed value of , we get a fixed line described with a single parameter . The surface is the union of these lines.
We use the following terminology:
- The line for each fixed value of is termed a ruling for the surface.
- The function is termed the ruled surface directrix or the base curve. For any ,
describes the position of the line. - The value
describes a direction vector along the line, and the function is termed a director curve.
Examples
| Ruled surface | Equational/implicit description | Functions and in a possible parametric description |
Comment |
|---|---|---|---|
| Euclidean plane |  (the  -plane) |
is the vector with coordinates  and is the vector  . |
The Euclidean plane is in fact a doubly ruled surface and also a minimal surface. |
| right circular cylinder (infinite version) |  (the right circular cylinder with base circle the unit circle in the -plane and axis along the  -axis). |
is the vector with coordinates  and is the vector  . |
|
| circular hyperboloid of one sheet |  |
This is in fact a doubly ruled surface | |
| elliptic hyperboloid of one sheet |  |
||
| hyperbolic paraboloid |  |
||
| helicoid |  |
it is the only ruled minimal surface other than the plane. |
