Ruled surface

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 (Fuchs & Tabachnikov 2007).

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 surface generated by two Bézier curves as directrices (red, green)

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

The directrix may collapse to a point (in case of a cone, see example below).

For the generation of a ruled surface by two directrices (or one directrix and the vectors of line directions) not only the geometric shape of these curves are essential but also the special parametric representations of them influence the shape of the ruled surface (see examples a), d)).

For any cone one can choose the apex as the directrix. This case shows: The directrix of a ruled surface may degenerate to a point.

Obviously the ruled surface is a doubly ruled surface, because any point lies on two lines of the surface.

For the considerations below any necessary derivative is assumed to exist.

The importance of this determinant condition shows the following statement:

The generators of any ruled surface coalesce with one family of its asymptotic lines. For developable surfaces they also form one family of its lines of curvature. It can be shown that any developable surface is a cone, a cylinder or a surface formed by all tangents of a space curve.[4]

The determinant condition for developable surfaces is used to determine numerically developable connections between space curves (directrices). The diagram shows a developable connection between two ellipses contained in different planes (one horizontal, the other vertical) and its development.[5]

An impression of the usage of developable surfaces in Computer Aided Design (CAD) is given in Interactive design of developable surfaces[6]

A historical survey on developable surfaces can be found in Developable Surfaces: Their History and Application[7]

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 through 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.

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.