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[parent] ruled surface (Topic)

A straight line $ g$ moving continuously in space sweeps a ruled surface. 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

$\displaystyle \vec{r} = \vec{a}(t), \quad \vec{r} = \vec{b}(t).$
Using two parameters $ s$ and $ t$ we express the position vector of an arbitrary point of the ruled surface as
$\displaystyle \vec{r} = \vec{a}(t)+ s \vec{b}(t).$
Here $ \vec{r} = \vec{a}(t)$ is a curve on the ruled surface and is called directrix or the base curve 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 directrix 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)

$\displaystyle \vec{r} = ct\vec{k}+ s (\vec{i}\cos{t}+\vec{j}\sin{t}) = \left(\!\begin{array}{c}s\cos{t}\ s\sin{t}\ ct\end{array}\!\right)\!.$
Figure: The helicoid as a ruled surface
\includegraphics{helicoid.eps}

2. The equation

$\displaystyle z = xy$
presents a hyperbolic paraboloid (if we rotate the coordinate system 45 degrees about the $ z$-axis using the formulae $ x = (x'-y')/\sqrt{2}$, $ y = (x'+y')/\sqrt{2}$, the equation gets the form $ x'^2-y'^2 = 2z$). Since the position vector of any point of the surface may be written using the parameters $ s$ and $ t$ as
$\displaystyle \vec{r} = \left(\!\begin{array}{c}0\ t\ 0\end{array}\!\right)\! +s\left(\!\begin{array}{c}1\ 0\ t\end{array}\!\right)\!,$
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.



"ruled surface" is owned by pahio. [ full author list (2) ]
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See Also: equation of plane, graph of equation $\,xy =$ constant

Also defines:  directrix, base curve, director curve, generatrix, generatrices, ruling, helicoid
Keywords:  surface, ruled

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Attachments:
cylinder (Topic) by stevecheng
developable surface (Topic) by pahio
cone in $\mathbb{R}^3$ (Definition) by pahio
generatrices of one-sheeted hyperboloid (Topic) by pahio
generatrices of hyperbolic paraboloid (Topic) by pahio
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Cross-references: one-sheeted hyperboloid, conical surfaces, plane, cylindrical surfaces, hyperbolic paraboloid, equation, circular helix, unit circle, position vector, parameters, parametric forms, curves, point, connected, surface, line, straight
There are 12 references to this entry.

This is version 13 of ruled surface, born on 2005-08-28, modified 2007-10-22.
Object id is 7347, canonical name is RuledSurface.
Accessed 6615 times total.

Classification:
AMS MSC: 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
 51M20 (Geometry :: Real and complex geometry :: Polyhedra and polytopes; regular figures, division of spaces)
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