A straight line moving continuously in space sweeps a ruled surface. Formally: A surface in is a ruled surface if it is connected and if for any point of , there is a line such that .
Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms
Using two parameters and we express the position vector of an arbitrary point of the ruled surface as
Here is a curve on the ruled surface and is called directrix or the base curve of the surface, while is the director curve of the surface. Every position of is a generatrix or ruling of the ruled surface.
1. Choosing the -axis (, ) as the directrix and the unit circle () as the director curve we get the helicoid (“screw surface”; cf. the circular helix)
2. The equation
presents a hyperbolic paraboloid (if we rotate the coordinate system 45 degrees about the -axis using the formulae , , the equation gets the form ). Since the position vector of any point of the surface may be written using the parameters and as
we see that it’s a question of a ruled surface with rectilinear directrix and director curve.
3. Other ruled surfaces are for example all cylindrical surfaces (plane included), conical surfaces, one-sheeted hyperboloid.