Ruled Surface

A Surface which can be swept out by a moving Line in space and therefore has a parameterization of the form

$${\bf x}(u,v)={\bf b}(u)+v\boldsymbol{\delta}(u),$$ (1)

where b is called the Directrix (also called the Base Curve) and $\boldsymbol{\delta}$ 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)

$$\left[{\matrix{a(cos u\mp v\sin u)\cr b(\sin u\pm \cos u)\cr... ...}\right]\pm v\left[{\matrix{-a\cos u\cr b\sin u\cr c}}\right],$$ (2)

the Hyperbolic Paraboloid (a doubly ruled surface)

$$\left[{\matrix{a(u+v)\cr \pm bv\cr u^2+2uv\cr}}\right] = \le... ... u^2\cr}}\right]+v\left[{\matrix{a\cr \pm b\cr 2u\cr}}\right],$$ (3)

Plücker's Conoid

$$\left[{\matrix{r\cos\theta\cr r\sin\theta\cr 2\cos\theta\sin... ...ht]+r\left[{\matrix{\cos\theta\cr \sin\theta\cr 0\cr}}\right],$$ (4)

and the Möbius Strip

$$a\left[{\matrix{\cos u+v\cos({\textstyle{1\over 2}}u)\cos u\... ...\over 2}}u)\sin u\cr \sin({\textstyle{1\over 2}}u)\cr}}\right]$$ (5)

(Gray 1993).

The only ruled Minimal Surfaces are the Plane and Helicoid (Catalan 1842, do Carmo 1986).

See also Asymptotic Curve, Cayley's Ruled Surface, Developable Surface, Director Curve, Directrix (Ruled Surface), Generalized Cone, Generalized Cylinder, Helicoid, Noncylindrical Ruled Surface, Plane, Right Conoid, Ruling

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/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 32-33, 1986.

Gray, A. ``Ruled Surfaces.'' Ch. 17 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 333-355, 1993.