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Ruled surface
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In geometry, a surface is ruled if through every point of there is a straight line that lies on . 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 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 is a curve lying in , and 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 can be parametrized as , where and are two non-intersecting curves lying on . In particular, when and move with constant speed along two skew lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet.
Doubly ruled surfaceA surface is doubly ruled if through every one of its points there are two distinct lines that lie on .
Particulars- The plane, which is also the only n-ally ruled surface for n = 3.
- The hyperbolic paraboloid
- The hyperboloid of one sheet
ApplicationDoubly ruled surfaces are used in the study of skew lines. Many hyperboloid structures have been built making use of only straight materials.
Developable surfaceA 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.
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