Ruled surface

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	$z = \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}}$
helicoid	$y = x \tan (z / c)$		it is the only ruled minimal surface other than the plane.

Relation with other properties

Stronger properties