“Cone”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cone

释义

Cone

A cone is a Pyramid with a circular Cross-Section. A right cone is a cone with its vertex above the center ofits base. A right cone of height can be described by the parametric equations

			(1)
			(2)
			(3)

for

and

. The Volume of a cone is therefore

(4)

where

is the base Area and

is the height. If the base is circular, then

(5)

This amazing fact was first discovered by Eudoxus,

and other proofs were subsequently found byArchimedes

in On the Sphere and Cylinder (ca. 225 BC

) and Euclid

in PropositionXII.10 of his Elements">Elements (Dunham 1990).

The Centroid can be obtained by setting in the equation for the centroid of theConical Frustum,

(6)

(Beyer 1987, p. 133) yielding

(7)

For a right circular cone, the Slant Height is

(8)

and the surface Area (not including the base) is

(9)

In discussions of Conic Sections, the word cone is often used to refer to two similar conesplaced apex to apex. This allows the Hyperbola to be defined as the intersection of a Plane with bothNappes (pieces) of the cone.

The Locus of the apex of a variable cone containing an Ellipse fixed in 3-space is a Hyperbola throughthe Foci of the Ellipse. In addition, the Locus of the apex of a cone containing thatHyperbola is the original Ellipse. Furthermore, the Eccentricities of theEllipse and Hyperbola are reciprocals.

See also Conic Section, Conical Frustum, Cylinder, Nappe, Pyramid, Sphere

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 129 and 133, 1987.

Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 76-77, 1990.

Yates, R. C. ``Cones.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 34-35, 1952.

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