Ruled surface

Ruled surface
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Ruled_surface".

In geometry, a surface $S$ is ruled if through every point of $S$ there is a straight line that lies on $S$ . The most familiar examples are the plane and the curved surface of a cylinder or cone. A ruled surface can be visualised as the surface formed by moving a "straight" line in space. For example, a cone is formed by keeping one end-point of a line fixed whilst moving the other end-point in a circle.

The properties of being ruled is preserved by projective maps, and therefore is a concept of projective geometry. Analogues for algebraic surfaces are studied in algebraic geometry.

A ruled surface $S$ can always be described (at least locally) as the set of points swept by a moving straight line, i.e. by a parametric equation of the form

$\mathbf{S}(t,u) = \mathbf{p}(t) + u \mathbf{r}(t)$

where $p$ is a curve lying in $S$ , and $r$ is a curve on the unit sphere. Thus, for example, if one uses

$\begin{align} \mathbf{p} &= (\cos(t), \sin(t), 0)\\ \mathbf{r} &= \left( \cos \left( \frac{t}{2} \right) \cos(t) , \cos \left( \frac{t}{2} \right) \sin(t), \sin \left( \frac{t}{2} \right) \right) \end{align}$

one obtains a ruled surface that contains the Möbius strip.

Alternatively, a ruled surface $S$ can be parametrized as $S (t, u) = (1 - u) p (t) + u q (t)$ , where $p$ and $q$ are two non-intersecting curves lying on $S$ . In particular, when $p (t)$ and $q (t)$ move with constant speed along two skew lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet.

A hyperboloid of one sheet. The wires are straight lines. Through any point on this surface pass two straight lines lying entirely on the surface, so it is doubly ruled.

content

1 Doubly ruled surface
- 1.1 Particulars
- 1.2 Application
2 Developable surface
3 References
4 External links

Doubly ruled surface

A surface $S$ is doubly ruled if through every one of its points there are two distinct lines that lie on $S$ .

Particulars

The plane, which is also the only n-ally ruled surface for n ≥ 3.
The hyperbolic paraboloid
The hyperboloid of one sheet

Application

Doubly ruled surfaces are used in the study of skew lines. Many hyperboloid structures have been built making use of only straight materials.

Developable surface

A developable surface — one that can be (locally) unrolled onto a flat plane without tearing or stretching — if complete, is necessarily ruled, but the converse is not always true. Thus the cylinder and cone are developable, but the general hyperboloid of one sheet is not.

References

External links

Picture of a hyperboloid of one sheet, at the University of Arizona. The strings in this picture are straight lines. This is a doubly ruled surface.

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