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Definition of a ruled surface: every point lies on a line
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.
Definition and parametric representation
Ruled surface generated by two Bézier curves as directrices (red, green)
A two dimensional differentiable manifold is called ruled surface, if it is the union of a one parametric family of lines. The lines of this family are the generators of the ruled surface.
Any curve with fixed parameter is a generator (line) and the curve is the directrix of the representation. The vectors describe the directions of the generators.
The directrix may collapse to a point (in case of a cone, see example below).
Alternatively the ruled surface (CR) can be described by
with the second directrix .
Alternatively, one can start with two non intersecting curves as directrices, and get by (CD) a ruled surface with line directions
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 theoretical investigations representation (CR) is more advantageous, because the parameter appears only once.
A simple calculation shows (see next section). Hence the given realization of a Möbius strip is not developable. But there exist developable Möbius strips.
Tangent planes, developable surfaces
For the considerations below any necessary derivative is supposed to exist.
For the determination of the normal vector at a point one needs the partial derivatives of the representation :
Hence the normal vector is
Because of (A mixed product with two equal vectors is always 0 !), vector is a tangent vector at any point . The tangent planes along this line are all the same, if is a multiple of . This is possible only, if the three vectors lie in a plane, i.e. they are linear dependent. The linear dependency of three vectors can be checked using the determinant of these vectors:
The tangent planes along the line are equal, if
The importance of this determinant condition shows the following statement:
A ruled surface is developable into a plane, if for any point the Gauss curvature vanishes. This is exactly the case if
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.
Developable connection of two ellipses and its development
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.
An impression of the usage of developable surfaces in Computer Aided Design (CAD) is given in Interactive design of developable surfaces
A historical survey on developable surfaces can be found in Developable Surfaces: Their History and Application
Ruled surfaces in algebraic geometry
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.
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.