What is "ruled surface"

Rule \Rule\, n. Syn: regulation; law; precept; maxim; guide; canon; order; method; direction; control; government; sway; empire. [1913 Webster] Rule \Rule\, v. t. [imp. & p. p. Ruled; p. pr. & vb. n. Ruling.] [Cf. OF. riuler, ruiler, L. regulare. See Rule, n., and cf. Regulate.]

1. To control the will and actions of; to exercise authority or dominion over; to govern; to manage.
   --Chaucer.

   A bishop then must be blameless; . . . one that ruleth well his own house, having his children in subjection.
   --1 Tim. iii. 2, 4.

2. To control or direct by influence, counsel, or persuasion; to guide; -- used chiefly in the passive.

   I think she will be ruled In all respects by me.
   --Shak.

3. To establish or settle by, or as by, a rule; to fix by universal or general consent, or by common practice.

   That's are ruled case with the schoolmen.
   --Atterbury.

4. (Law) To require or command by rule; to give as a direction or order of court.

5. To mark with lines made with a pen, pencil, etc., guided by a rule or ruler; to print or mark with lines by means of a rule or other contrivance effecting a similar result; as, to rule a sheet of paper of a blank book.

   Ruled surface (Geom.), any surface that may be described by a straight line moving according to a given law; -- called also a scroll.

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. The most familiar examples (illustrated here in three-dimensional Euclidean space) are the plane and the curved surface of a cylinder or cone. Other examples are 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 always be described (at least locally) 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.