Hyperbola

For hyperbole, the figure of speech, see hyperbole.

In mathematics, a hyperbola is a type of conic section (literally: 'exaggeration' from the Greek word ’υπερβολή).

• Geometrically, it is defined as the intersection between a cone and a plane which cuts through both halves of the cone.

• Analytically, it is defined as the set of all points for which the difference in the distance to two fixed points (called the foci) is constant.

• It can also be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola. These foci lie on the transverse axis and their midpoint is called the center.

A hyperbola has the property that a ray originating at one of the foci is reflected in such a way as to appear to have originated at the other focus.

A special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angles. The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy=c, where c is a constant.

Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.

A body that has sufficient energy to escape the gravitational field of a massive body moves in a hyperbolic trajectory with the massive body at one of the foci.

Equations

Cartesian

The eccentricity is given by
:

Polar

:
:
:
:

Parametric

:
:

See also

Ellipse

parabola

conic section

Dandelin spheres

hyperbole

hyperbolic sector

hyperbolic angle

Inverted World (Book)

External links

Mathworld - Hyperbola