Difference between revisions of "Ruled surface"
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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> || || | ||
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− | | [[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]] |
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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. |
Latest revision as of 14:50, 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 ![]() ![]() |
Comment |
---|---|---|---|
Euclidean plane | ![]() ![]() |
![]() ![]() ![]() ![]() |
The Euclidean plane is in fact a doubly ruled surface and also a minimal surface. |
right circular cylinder (infinite version) | ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
|
circular hyperboloid of one sheet | ![]() |
This is in fact a doubly ruled surface | |
elliptic hyperboloid of one sheet | ![]() |
||
hyperbolic paraboloid | ![]() |
This is in fact a doubly ruled surface | |
helicoid | ![]() |
it is the only ruled minimal surface other than the plane. |