ruled surface

A straight line g moving continuously in space sweeps a ruled surfaceMathworldPlanetmath.  Formally:  A surface S in 3 is a ruled surface if it is connected and if for any point p of S, there is a line g such that  pgS.

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 ( 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.


1.  Choosing the z-axis (r=ctk,  c0) as the and the unit circle (r=icost+jsint) as the director curve we get the helicoid (“screw surface”; cf. the circular helix)


2.  The equation


presents a hyperbolic paraboloidMathworldPlanetmath (if we rotate the coordinate systemMathworldPlanetmath ( 45 about the z-axis using the formulae  x=(x-y)/2,  y=(x+y)/2,  the equation gets the form  x2-y2=2z).  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 directrixPlanetmathPlanetmath and director curve.

3.  Other ruled surfaces are for example all cylindrical surfacesMathworldPlanetmath (plane included), conical surfaces, one-sheeted hyperboloid (

Title ruled surface
Canonical name RuledSurface
Date of creation 2016-03-03 17:28:55
Last modified on 2016-03-03 17:28:55
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Topic
Classification msc 51M20
Classification msc 51M04
Related topic EquationOfPlane
Related topic GraphOfEquationXyConstant
Defines directrix
Defines base curve
Defines director curve
Defines generatrix
Defines generatrices
Defines ruling
Defines helicoid