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Concept
Ruled surface
Summary
In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.
A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points .
The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geometry, ruled surfaces are sometimes considered to be surfaces in affine or projective space over a field, but they are also sometimes considered as abstract algebraic surfaces without an embedding into affine or projective space, in which case "straight line" is understood to mean an affine or projective line.
A two dimensional differentiable manifold is called a ruled surface if it is the union of one parametric family of lines. The lines of this family are the generators of the ruled surface.
A ruled surface can be described by a parametric representation of the form
(CR) .
Any curve with fixed parameter is a generator (line) and the curve is the directrix of the representation. The vectors describe the directions of the generators.
The directrix may collapse to a point (in case of a cone, see example below).
Alternatively the ruled surface (CR) can be described by
(CD)
with the second directrix .
Official source
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