Ruled surface

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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

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

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) + uq(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.

[edit] 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.

[edit] Particulars

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

[edit] Application

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

[edit] Developable surface

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.

[edit] References

• Eric W. Weisstein, Developable Surface at MathWorld.
• Examples of developable surfaces on the Rhino3DE website

[edit] 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.