Ruled surface

This article defines a property that makes sense for a surface embedded in $\mathbb {R} ^{3}$, 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 $\mathbb {R} ^{3}$ 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 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 $xy$-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	$z={\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}$		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

• Doubly ruled surface
• Developable surface