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

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 curve on the unit-radius 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 straight 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 — 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 surfaceIn mathematics, a minimal surface is a surface with mean curvature of zero, or, equivalently, a surface of minimum area subject to constraints on the location of its boundary. Examples of minimal surfaces include catenoids and helicoids. A soap film strets 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 geometryIn a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the name projective geometry was a stepping stone from analytic geometry to algebraic geometry. When treated in terms of hom. Analogues for algebraic surfaceIn mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the complex number field, an algebraic surface is therefore of complex dimension two (as a complex manifold) and so of dimension four as a smooth ms are studied in algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations . When there is more than o.