Ruled surface

From Wikipedia, the free encyclopedia

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. The last two are special cases of ruled quadrics (which also include the hyperbolic paraboloid, the hyperboloid of one sheet, and the conical surface with elliptical directrix). Other examples are the right conoid, the helicoid, and the tangent developable of a smooth curve in space.

An elliptic hyperboloid of one sheet. The wires are straight lines. Two straight lines lying entirely on this surface pass through each point on the surface, so it is doubly ruled.

A surface $S$ is doubly ruled if through every one of its points there are two distinct lines that lie on $S$ . The plane, the hyperbolic paraboloid, and the hyperboloid of one sheet are the only doubly-ruled quadrics.

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

$S(t,u) = p(t) + u r(t) \,\!$

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

$p(t) = (\cos(t), \sin(t), 0) \,\!$

$r(t) = (\cos (t/2) \cos(t) , \cos (t/2) \sin(t), \sin (t/2)) \,\!$

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 developable surface — one that can be (locally) unrolled onto a flat plane without tearing or stretching — if complete, it 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. The only minimal surfaces that are ruled are the plane and the helicoid.

The properties of being ruled or doubly-ruled are preserved by projective maps, and therefore are concepts of projective geometry. Analogues for algebraic surfaces are studied in algebraic geometry.