Hyperbola

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.

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