In geometry, a surfaceS is ruled 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 ellipticaldirectrix, 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. 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.
Any curve with fixed parameter is a generator and the curve is the directrix of the representation. The vectors describe the directions of the generators. The directrix may collapse to a point. Alternatively the ruled surface ' can be described by
'
with the second directrix. Alternatively, one can start with two non intersecting curves as directrices, and get by ' a ruled surface with line directions For the generation of a ruled surface by two directrices 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 ). For theoretical investigations representation is more advantageous, because the parameter appears only once.
Examples
a) Right circular cylinder: with b) Right circular cone: with In this case one could have used the apex as the directrix, i.e.: and as the line directions. 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. c) Helicoid: The directrix is the z-axis, the line directions are and the second directrix is a helix. The helicoid is a special case of the ruled generalized helicoids. d) Cylinder, cone and hyperboloids: The parametric representation has two horizontal circles as directrices. The additional parameter allows to vary the parametric representations of the circles. For A hyperboloid of one sheet is a doubly ruled surface. e) Hyperbolic paraboloid: If the two directrices in are the lines one gets which is the hyperbolic paraboloid that interpolates the 4 points bilinearly. Obviously the ruled surface is a doubly ruled surface, because any point lies on two lines of the surface. For the example shown in the diagram: The hyperbolic paraboloid has the equation. f) Möbius strip: The ruled surface with contains a Möbius strip. The diagram shows the Möbius strip for. A simple calculation shows . 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 , 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.
The determinant condition for developable surfaces is used to determine numerically developable connections between space curves. The diagram shows a developable connection between two ellipses contained in different planes and its development. An impression of the usage of developable surfaces in Computer Aided Design 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. 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
Doubly ruled surfaces are the inspiration for curved hyperboloid structures that can be built with a latticework of straight elements, namely:
Hyperbolic paraboloids, such as saddle roofs.
Hyperboloids of one sheet, such as cooling towers and some trash bins.
The RM-81 Agenarocket engine employed straight cooling channels that were laid out in a ruled surface to form the throat of the nozzle section.
