Hyperbola（双曲线）

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A hyperbola (plural "hyperbolas"; Gray 1997, p. 45) is a conic section defined as the locus of all points  in the plane the difference of whose distances  and  from two fixed points (the foci  and ) separated by a distance  is a given positive constant ,

(1)

(Hilbert and Cohn-Vossen 1999, p. 3). Letting  fall on the left -intercept requires that

(2)

so the constant is given by , i.e., the distance between the -intercepts (left figure above). The hyperbola has the important property that a ray originating at a focus  reflects in such a way that the outgoing path lies along the line from the other focus through the point of intersection (right figure above).

The special case of the rectangular hyperbola, corresponding to a hyperbola with eccentricity , was first studied by Menaechmus. Euclid and Aristaeus wrote about the general hyperbola, but only studied one branch of it. The hyperbola was given its present name by Apollonius, who was the first to study both branches. The focus and conic section directrix were considered by Pappus (MacTutor Archive). The hyperbola is the shape of an orbit of a body on an escape trajectory (i.e., a body with positive energy), such as some comets, about a fixed mass, such as the sun.

The hyperbola can be constructed by connecting the free end  of a rigid bar , where  is a focus, and the other focus  with a string . As the bar  is rotated about  and  is kept taut against the bar (i.e., lies on the bar), the locus of  is one branch of a hyperbola (left figure above; Wells 1991). A theorem of Apollonius states that for a line segment tangent to the hyperbola at a point  and intersecting the asymptotes at points  and , then  is constant, and  (right figure above; Wells 1991).

Let the point  on the hyperbola have Cartesian coordinates , then the definition of the hyperbola  gives

(3)

Rearranging and completing the square gives

(4)

and dividing both sides by  results in

(5)

By analogy with the definition of the ellipse, define

(6)

so the equation for a hyperbola with semimajor axis  parallel to the x-axis and semiminor axis  parallel to the y-axis is given by

(7)

or, for a center at the point  instead of ,

(8)

Unlike the ellipse, no points of the hyperbola actually lie on the semiminor axis, but rather the ratio  determines the vertical scaling of the hyperbola. The eccentricity  of the hyperbola (which always satisfies ) is then defined as

(9)

In the standard equation of the hyperbola, the center is located at , the foci are at , and the vertices are at . The so-called asymptotes(shown as the dashed lines in the above figures) can be found by substituting 0 for the 1 on the right side of the general equation (8),

(10)

and therefore have slopes .

The special case  (the left diagram above) is known as a rectangular hyperbola because the asymptotes are perpendicular.

The hyperbola can also be defined as the locus of points whose distance from the focus  is proportional to the horizontal distance from a vertical line  known as the conic section directrix, where the ratio is . Letting  be the ratio and  the distance from the center at which the directrix lies, then

			(11)
			(12)

where  is therefore simply the eccentricity .

Like noncircular ellipses, hyperbolas have two distinct foci and two associated conic section directrices, each conic section directrix being perpendicular to the line joining the two foci (Eves 1965, p. 275).

The focal parameter of the hyperbola is

			(13)
			(14)
			(15)

In polar coordinates, the equation of a hyperbola centered at the origin (i.e., with ) is

(16)

In polar coordinates centered at a focus,

(17)

as illustrated above.

The two-center bipolar coordinates equation with origin at a focus is

(18)

Parametric equations for the right branch of a hyperbola are given by

			(19)
			(20)

where  is the hyperbolic cosine and  is the hyperbolic sine, which ranges over the right branch of the hyperbola.

A parametric representation which ranges over both branches of the hyperbola is

			(21)
			(22)

with  and discontinuities at . The arc length, curvature, and tangential angle for the above parametrization are

			(23)
			(24)
			(25)

where  is an elliptic integral of the second kind.

The special affine curvature of the hyperbola is

(26)

The locus of the apex of a variable cone containing an ellipse fixed in three-space is a hyperbola through the foci of the ellipse. In addition, the locus of the apex of a cone containing that hyperbola is the original ellipse. Furthermore, the eccentricities of the ellipse and hyperbola are reciprocals.

REFERENCES:

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 199-200 and 218, 1987.

Casey, J. "The Hyperbola." Ch. 7 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 250-284, 1893.

Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 75-76, 1996.

Coxeter, H. S. M. "Conics" §8.4 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115-119, 1969.

Eves, H. A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, 1965.

Fukagawa, H. and Pedoe, D. "The One Hyperbola." §5.2 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 51 and 136-138, 1989.

Gardner, M. "Hyperbolas." Ch. 15 in Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 205-218, 1989.

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, pp. 3-4, 1999.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 79-82, 1972.

Lockwood, E. H. "The Hyperbola." Ch. 3 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 24-33, 1967.

Loomis, E. S. "The Hyperbola." §2.3 in The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, pp. 22-23, 1968.

MacTutor History of Mathematics Archive. "Hyperbola." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hyperbola.html.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 106-109, 1991.

Yates, R. C. "Conics." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 36-56, 1952.

http://mathworld.wolfram.com/Hyperbola.html

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