ruled surface

A straight line $g$ moving continuously in space sweeps a .  Formally:  A surface $S$ in $\mathbb{R}^{3}$ 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

 $\vec{r}\;=\;\vec{a}(t),\qquad\vec{r}\;=\;\vec{b}(t).$

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

 $\vec{r}\;=\;\vec{a}(t)+s\,\vec{b}(t).$

Here  $\vec{r}=\vec{a}(t)$  is a curve on the ruled surface and is called or the of the surface, while  $\vec{r}=\vec{b}(t)$  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 ($\vec{r}=ct\vec{k}$,  $c\neq 0$) as the and the unit circle ($\vec{r}=\vec{i}\cos{t}+\vec{j}\sin{t}$) as the director curve we get the helicoid (“screw surface”; cf. the circular helix)

 $\vec{r}\;=\;ct\vec{k}+s\,(\vec{i}\cos{t}+\vec{j}\sin{t})\;=\;\left(\!\begin{% array}[]{c}s\,\cos{t}\\ s\,\sin{t}\\ ct\end{array}\!\right)\!.$

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

 $\vec{r}\;=\;\left(\!\begin{array}[]{c}0\\ t\\ 0\end{array}\!\right)\!+s\left(\!\begin{array}[]{c}1\\ 0\\ t\end{array}\!\right)\!,$

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