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 (http://planetmath.org/PositionVector) of an arbitrary point of the ruled surface as
Here is a curve on the ruled surface and is called or the 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 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 (http://planetmath.org/RotationMatrix) 45 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 (http://planetmath.org/QuadraticSurfaces).
|Date of creation||2016-03-03 17:28:55|
|Last modified on||2016-03-03 17:28:55|
|Last modified by||pahio (2872)|