ruled surface (original) (raw)

A straight line g moving continuously in space sweeps a ruled surface. Formally: A surface S in ℝ3 is a ruled surface if it is connected and if for any point p of S, there is a line g such that p∈g⊂S.

Such a surface may be formed by using two auxiliary curves given e.g. in the parametric forms

Using two parameters s and t we express the position vector (http://planetmath.org/PositionVector) of an arbitrary point of the ruled surface as

Here r→=a→⁢(t) is a curve on the ruled surface and is called or the of the surface, while r→=b→⁢(t) is the director curve of the surface. Every position of g is a generatrix or ruling of the ruled surface.

Examples

1. Choosing the z-axis (r→=c⁢t⁢k→, c≠0) as the and the unit circle (r→=i→⁢cos⁡t+j→⁢sin⁡t) 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 z-axis using the formulae x=(x′-y′)/2, y=(x′+y′)/2, the equation gets the form x′⁣2-y′⁣2=2⁢z). Since the position vector of any point of the surface may be written using the parameters s and t 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).