A ruled surface is a surface that can be swept out by moving a line in space. It therefore has a parameterization
of the form
 |
(1)
|
where 
is called the ruled surface directrix (also
called the base curve) and 
is the director curve.
The straight lines themselves are called rulings. The
rulings of a ruled surface are asymptotic curves.
Furthermore, the Gaussian curvature on a ruled
regular surface is everywhere nonpositive.
Examples of ruled surfaces include the elliptic hyperboloid
of one sheet (a doubly ruled surface)
![[a(cosu∓vsinu); b(sinu+/-vcosu); +/-cv]=[acosu; bsinu; 0]+/-v[-asinu; bcosu; c],](/images/equations/RuledSurface/NumberedEquation2.svg) |
(2)
|
the hyperbolic paraboloid (a doubly
ruled surface)
![[a(u+v); +/-bv; u^2+2uv]=[au; 0; u^2]+v[a; +/-b; 2u],](/images/equations/RuledSurface/NumberedEquation3.svg) |
(3)
|
Plücker's conoid
![[rcostheta; rsintheta; 2costhetasintheta]=[0; 0; 2costhetasintheta]+r[costheta; sintheta; 0],](/images/equations/RuledSurface/NumberedEquation4.svg) |
(4)
|
and the Möbius strip
![a[cosu+vcos(1/2u)cosu; sinu+vcos(1/2u)sinu; vsin(1/2u)]=a[cosu; sinu; 0]+av[cos(1/2u)cosu; cos(1/2u)sinu; sin(1/2u)]](/images/equations/RuledSurface/NumberedEquation5.svg) |
(5)
|
(Gray 1993).
The only ruled minimal surfaces are the plane
and helicoid (Catalan 1842, do Carmo 1986).
See also
Asymptotic Curve,
Cayley Cubic,
Developable Surface,
Director
Curve,
Doubly Ruled Surface,
Generalized
Cone,
Generalized Cylinder,
Helicoid,
Noncylindrical Ruled Surface,
Plane,
Right Conoid,
Ruled
Surface Directrix,
Ruling
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References
Catalan E. "Sur les surfaces réglées dont l'aire est un minimum." J. Math. Pure. Appl. 7, 203-211,
1842.do Carmo, M. P. "The Helicoid." §3.5B in Mathematical
Models from the Collections of Universities and Museums (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, pp. 44-45, 1986.Fischer, G. (Ed.).
Plates 32-33 in Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig,
Germany: Vieweg, pp. 32-33, 1986.Gray, A. "Ruled Surfaces."
Ch. 19 in Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, pp. 431-456, 1993.Hilbert, D. and Cohn-Vossen,
S. Geometry
and the Imagination. New York: Chelsea, p. 15, 1999.Steinhaus,
H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 242-243, 1999.Referenced
on Wolfram|Alpha
Ruled Surface
Cite this as:
Weisstein, Eric W. "Ruled Surface." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RuledSurface.html
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