“Hyperboloid”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hyperboloid

释义

Hyperboloid

A Quadratic Surface which may be one- or two-sheeted.

The one-sheeted circular hyperboloidis a doubly Ruled Surface. When oriented along the z-Axis, the one-sheeted circular hyperboloidhas Cartesian Coordinates equation

(1)

and parametric equation

			(2)
			(3)
			(4)

for

(left figure). Other parameterizations include

			(5)
			(6)
			(7)

(middle figure), or

			(8)
			(9)
			(10)

(right figure). An obvious generalization gives the one-sheeted Elliptic Hyperboloid.

A two-sheeted circular hyperboloid oriented along the z-Axis has Cartesian Coordinates equation

(11)

The parametric equations are

			(12)
			(13)
			(14)

for

. Note that the plus and minus signs in

correspond to the upper and lower sheets. The two-sheetedcircular hyperboloid oriented along the x-Axis has Cartesian equation

(15)

and parametric equations

			(16)
			(17)
			(18)

(Gray 1993, p. 313). Again, an obvious generalization gives the two-sheeted Elliptic Hyperboloid.

The Support Function of the hyperboloid of one sheet

(19)

(20)

and the Gaussian Curvature is

(21)

The Support Function of the hyperboloid of two sheets

(22)

(23)

and the Gaussian Curvature is

(24)

(Gray 1993, pp. 296-297).

See also Catenoid, Ellipsoid, Elliptic Hyperboloid,Hyperboloid Embedding, Paraboloid, Ruled Surface

References

Fischer, G. (Ed.). Plates 67 and 69 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 62 and 64, 1986.

Gray, A. ``The Hyperboloid of Revolution.'' §18.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 296-297, 311-314, and 369-370, 1993.

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