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)

(2)

the hyperbolic paraboloid (a doubly ruled surface)

(3)

Plücker's conoid

(4)

and the Möbius strip

(5)

(Gray 1997).

The only ruled minimal surfaces are the plane and helicoid (Catalan 1842, do Carmo 1986).

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.

CITE THIS AS:

Weisstein, Eric W. "Ruled Surface." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RuledSurface.html