Ruled surface

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A hyperboloid of one sheet is a doubly ruled surface: it can be generated by either of two families of straight lines.

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.

Ruled surfaces in differential geometry[edit]

Parametric representation[edit]

A ruled helicoid

The "moving line" view means that a ruled surface has a parametric representation of the form

S(t,u) = p(t) + u r(t)\

where S(t,u) is the generic point on the surface, p(t) is point that traces a curve lying on the surface, and r(t) is a unit-length vector that traces a curve on the unit sphere. Thus, for example, if one uses


\begin{align}
p(t) &= (\cos(2t), \sin(2t), 0)\\
r(t) &= ( \cos t \cos 2 t , \cos t \sin 2 t, \sin t )
\end{align}

one obtains a ruled surface that contains the Möbius strip.

Alternatively, a ruled surface can be parametrized as S(t,u) = (1-u) p(t) + u q(t), where p and q are two non-intersecting curves lying on the surface. In particular, when p(t) and q(t) move with constant speed along two skew lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet.

Developable surface[edit]

A developable surface is a surface that can be (locally) unrolled onto a flat plane without tearing or stretching it. If a developable surface lies in three-dimensional Euclidean space, and is complete, then it is necessarily ruled, but the converse is not always true. For instance, the cylinder and cone are developable, but the general hyperboloid of one sheet is not. More generally, any developable surface in three dimensions is part of a complete ruled surface, and so itself must be locally ruled. There are developable surfaces embedded in four dimensions which are however not ruled. (Hilbert & Cohn-Vossen 1952, pp. 341–342)

Ruled surfaces in algebraic geometry[edit]

A doubly ruled hyperbolic paraboloid with equation z=xy

In algebraic geometry, ruled surfaces were originally defined as projective surfaces in projective space containing a straight line through any given point. This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces are defined to be abstract projective surfaces satisfying this condition that there is a projective line though any point. This is equivalent to saying that they are birational to the product of a curve and a projective line. Sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a fibration over a curve with fibers that are projective lines. This excludes the projective plane, which has a projective line though every point but cannot be written as such a fibration.

Ruled surfaces appear in the Enriques classification of projective complex surfaces, because every algebraic surface of Kodaira dimension −∞ is a ruled surface (or a projective plane, if one uses the restrictive definition of ruled surface). Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve. The ruled surfaces with base curve of genus 0 are the Hirzebruch surfaces.

Ruled surfaces in architecture[edit]

Doubly ruled surfaces are the inspiration for curved hyperboloid structures that can be built with a latticework of straight elements, namely:

The RM-81 Agena rocket engine employed straight cooling channels that were laid out in a ruled surface to form the throat of the nozzle section.

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