Ruled surface: Difference between revisions

Latest revision as of 14:50, 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:

$x (u, v) = b (u) + v δ (u)$

where $b$ and $δ$ 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 $b$ is termed the ruled surface directrix or the base curve. For any $u$ , $b (u)$ describes the position of the line.
The value $δ (u)$ describes a direction vector along the line, and the function $δ$ is termed a director curve.

Examples

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

Relation with other properties

Stronger properties

@@ 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.
+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:
+<math>\! \mathbf{x}(u,v) = \mathbf{b}(u) + v \delta(u)</math>
+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.
+We use the following terminology:
+* The line for each fixed value of <math>u</math> is termed a ''ruling'' for the surface.
+* 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.
+* The value <math>\delta(u)</math> describes a ''direction'' vector along the line, and the function <math>\delta</math> is termed a '''director curve'''.
+==Examples==
+{| class="sortable" border="1"
+! 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]] 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>. ||
+|-
+| [[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]]
+|-
+| [[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> || || 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.
+|}
 ==Relation with other properties==
@@ Line 9: / Line 43: @@
 ===Stronger properties===
-* [[Doubly ruled surface]]
+* [[Weaker than::Doubly ruled surface]]
-* [[Developable surface]]
+* [[Weaker than::Developable surface]]