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 equal angles 
in the right figure are measured between the rays and
the tangent line at 
.
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.
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|
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 blue lines in the figure above) 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 
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 
. For the hyperbolic parametrization 
, 
with 
,
the arc length, curvature,
and tangential angle are
 |  |  |
(23)
|
 |  |  |
(24)
|
 |  |  |
(25)
|
where 
is an elliptic
integral of the second kind with elliptic modulus 
. The arc length is measured
from 
, so the length of the segment from 
to 
, where 
,
is 
.
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.
