In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The plane, cylinder, and cone are the most familiar examples. The first two are special cases of ruled quadrics (which also include the hyperbolic paraboloid, the hyperboloid of one sheet, and the conical surface with elliptical directrix). Other examples are the right conoid, the helicoid, and the tangent developable of a smooth curve in space. Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... In mathematics, a surface is a two-dimensional manifold. ... In mathematics, a plane is the fundamental two-dimensional object. ... The word cylinder has several meanings. ... A cone is a basic geometrical shape: see cone (geometry). ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or (hyperbolic paraboloid). ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation:  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. ... In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ... The helicoid is one of the first minimal surfaces discovered. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

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

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

S(t,u) = p(t) + ur(t)

where p is a curve lying in S, and r is curve on the unit-radius sphere. Thus, for example, if one uses A sphere is, roughly speaking, a ball-shaped object. ...

p(t) = (cost,sint,0)

r(t) = (cos(t / 2)cost,cos(t / 2)sint,sin(t / 2))

one obtains a ruled surface that contains the Möbius strip. A Möbius strip made with a piece of paper and tape. ...

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 straight lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet. In telecommunication, the term skew has the following meanings: 1. ...

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 surfaces that are ruled are the plane and the helicoid. Verrill Minimal Surface In mathematics, a minimal surface is a surface with a mean curvature of zero. ...

The properties of being ruled or doubly-ruled are preserved by projective maps, and therefore are concepts of projective geometry. Analogues for algebraic surfaces are studied in algebraic geometry. A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ... Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...