ruled surface

A straight line $g$ moving continuously in space sweeps a ruled surface.  Formally:  A surface $S$ in ${ℝ}^3$ is a ruled surface if it is connected and if for any point $p$ of $S$, there is a line $g$ such that  $p\in g\subset S$.

Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms

$$\overrightarrow{r}=\overrightarrow{a}(t),\overrightarrow{r}=\overrightarrow{b}(t).$$

Using two parameters $s$ and $t$ we express the position vector (http://planetmath.org/PositionVector) of an arbitrary point of the ruled surface as

$$\overrightarrow{r}=\overrightarrow{a}(t)+s\overrightarrow{b}(t).$$

Here  $\overrightarrow{r}=\overrightarrow{a}(t)$  is a curve on the ruled surface and is called or the of the surface, while  $\overrightarrow{r}=\overrightarrow{b}(t)$  is the director curve of the surface.  Every position of $g$ is a generatrix or ruling of the ruled surface.

Examples

1.  Choosing the $z$-axis ($\overrightarrow{r}=ct\overrightarrow{k}$,  $c\ne 0$) as the and the unit circle ($\overrightarrow{r}=\overrightarrow{i}\cos t+\overrightarrow{j}\sin t$) as the director curve we get the helicoid (“screw surface”; cf. the circular helix)

$$\overrightarrow{r}=ct\overrightarrow{k}+s(\overrightarrow{i}\cos t+\overrightarrow{j}\sin t)=\left(\begin{array}{c}\hfill s\cos t\hfill \\ \hfill s\sin t\hfill \\ \hfill ct\hfill \end{array}\right).$$

2.  The equation

$$z=xy$$

presents a hyperbolic paraboloid (if we rotate the coordinate system (http://planetmath.org/RotationMatrix) 45 about the $z$-axis using the formulae  $x=(x^{\prime }-y^{\prime })/\sqrt{2}$,  $y=(x^{\prime }+y^{\prime })/\sqrt{2}$,  the equation gets the form  $x^{\prime 2}-y^{\prime 2}=2z$).  Since the position vector of any point of the surface may be written using the parameters $s$ and $t$ as

$$\overrightarrow{r}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill t\hfill \\ \hfill 0\hfill \end{array}\right)+s\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill t\hfill \end{array}\right),$$

we see that it’s a question of a ruled surface with rectilinear directrix and director curve.

3.  Other ruled surfaces are for example all cylindrical surfaces (plane included), conical surfaces, one-sheeted hyperboloid (http://planetmath.org/QuadraticSurfaces).

Title	ruled surface
Canonical name	RuledSurface
Date of creation	2016-03-03 17:28:55
Last modified on	2016-03-03 17:28:55
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	19
Author	pahio (2872)
Entry type	Topic
Classification	msc 51M20
Classification	msc 51M04
Related topic	EquationOfPlane
Related topic	GraphOfEquationXyConstant
Defines	directrix
Defines	base curve
Defines	director curve
Defines	generatrix
Defines	generatrices
Defines	ruling
Defines	helicoid