“Helicoid”的概念、定义、习题解题方法及翻译-数学辞典

单词

Helicoid

释义

Helicoid

The Minimal Surface having a Helix as its boundary. It is the only Ruled MinimalSurface other than the Plane (Catalan 1842, do Carmo 1986). For many years, the helicoid remained the only knownexample of a complete embedded Minimal Surface of finite topology with infinite Curvature. However, in 1992 asecond example, known as Hoffman's Minimal Surface and consisting of a helicoid with a Hole, was discovered (Sci. News 1992).

The equation of a helicoid in Cylindrical Coordinates is

(1)

In Cartesian Coordinates, it is

(2)

It can be given in parametric form by

			(3)
			(4)
			(5)

which has an obvious generalization to the Elliptic Helicoid. The differentials are

			(6)
			(7)
			(8)

so the Line Element on the surface is




			(9)

and the Metric components are

			(10)
			(11)
			(12)

From Gauss's Theorema Egregium, the Gaussian Curvature is then

(13)

The Mean Curvature is

(14)

and the equation for the Lines of Curvature is

(15)

The helicoid can be continuously deformed into a Catenoid by the transformation

			(16)
			(17)
			(18)

where

corresponds to a helicoid and

to a Catenoid.

If a twisted curve (i.e., one with Torsion ) rotates about afixed axis and, at the same time, is displaced parallel to such that the speed of displacement is alwaysproportional to the angular velocity of rotation, then generates a Generalized Helicoid.

See also Calculus of Variations, Catenoid, Elliptic Helicoid, Generalized Helicoid, Helix,Hoffman's Minimal Surface, Minimal Surface

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.). Plate 91 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 87, 1986.

Geometry Center. ``The Helicoid.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/helicoid/.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 264, 1993.

Kreyszig, E. Differential Geometry. New York: Dover, p. 88, 1991.

Meusnier, J. B. ``Mémoire sur la courbure des surfaces.'' Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.

Peterson, I. ``Three Bites in a Doughnut.'' Sci. News 127, 168, Mar. 16, 1985.

``Putting a Handle on a Minimal Helicoid.'' Sci. News 142, 276, Oct. 24, 1992.

Wolfram, S. The Mathematica Book, 3rd ed. Champaign, IL: Wolfram Media, p. 164, 1996.

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