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Online lecture:

Differential characterization: .
Cartesian parametrization: , union of the generatrices D passing by  with direction vector  (more simply: ); directrix cone: .

The distribution parameter d of the generatrix passing by M1 is defined by .
and the surface is said to be right-handed if  and left-handed if  ; it is developable iff = 0.
The distribution parameter can be geometrically interpreted as the number  where  is the distance between D and Du+du  and  is the angle between D and Du+du  .
See another interpretation of this parameter, as well as the formula giving the curvature, at striction line.

A ruled surface is a surface that is the union of lines, called its generatrices. A directrix cone is associated to it, union of the lines passing by a given point and parallel to the generatrices.

By three curves, there passes in general a unique ruled surface, union of the lines meeting these three curves. If the three curves are algebraic of respective degrees p,q,r, then the surface is "in general" algebraic of degree 2pqr.
The family of lines supported on two given curves does not generate, in general, a surface; but it is the case if we add another condition, like: The examples below show various ruled surfaces supported on a semicircle and a half-ellipse:
1) the lines have to meet a third given curve (hence family of the lines supported on three curves)

the third curve is a line
2) they must be parallel to a given "directrix" plane (Catalan surface)
3) the surface is developable (hence the envelope of the family of common tangent planes of the two curves, such planes being obtained by taking a tangent to a curve, and a tangent to the other curve, secant to the first tangent)
4) the distance between two contact points is constant (see the examples of the milk carton and the oloid)

The points of a ruled surface are hyperbolic or parabolic; when a point is parabolic, all the points on its generatrix are also parabolic; this happens when it is tangent to the striction line and corresponds to the case where the distribution parameter is equal to zero.

 - the cones, the cylinders, and more generally, the developable surfaces (case where all the generatrices are parabolic).
 - the conoids (including the hyperbolic paraboloid) and more generally the Catalan surfaces (planar directrix cone) or the conoidal surfaces (including the milk carton).
 - the one-sheeted hyperboloids.
 - the right helicoids (including the right helicoid and the developable helicoid).
- the Möbius surface and more generally the Cayley ruled cubics.
- Hector Guimard's surface.
- examples of Seifert surfaces.

The ruled non-degenerate quadrics are the surfaces union of lines meeting three lines two by two non coplanar: hyperbolic paraboloid in the case where the three lines are parallel to a fixed plane and one-sheeted hyperboloid in the other case; they are the only doubly ruled surfaces (i.e. that are the union of two distinct families of lines).

The cubic ruled surfaces are the cones and the cylinders with a cubic as a directrix, the conoidal surfaces of degree 3 and the Cayley ruled surfaces.

The ruled surfaces of revolution are the one-sheeted hyperboloids.

See also their cousins, the circled surfaces.

Fences at La Villette, Paris (2014)

Catenaries forming elegant ruled surfaces

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© Robert FERRÉOL, Robert March 2017