# Dictionary Definition

hyperbolic adj

1 enlarged beyond truth or reasonableness; "had
an exaggerated (or inflated) opinion of himself"; "a hyperbolic
style" [syn: exaggerated, inflated]

2 of or relating to a hyperbola; "hyperbolic
functions"

# User Contributed Dictionary

## English

### Adjective

- of, or relating to hyperbole; exaggerated
- mathematics notcomparable of or pertaining to a hyperbola, or to the hyperbolic function

#### Derived terms

- hyperbolic 3-manifold
- hyperbolic algebra
- hyperbolic angle
- hyperbolic automorphism
- hyperbolic coordinate
- hyperbolic cosecant
- hyperbolic cosine
- hyperbolic cosine integral
- hyperbolic cotangent
- hyperbolic cube
- hyperbolic cylinder
- hyperbolic differential equation
- hyperbolic discounting
- hyperbolic disk
- hyperbolic dodecahedron
- hyperbolic fixed point
- hyperbolic function
- hyperbolic geometry
- hyperbolic group
- hyperbolic growth
- hyperbolic helicoid
- hyperbolic icosahedron
- hyperbolic inverse function
- hyperbolic lemniscate function
- hyperbolic link
- hyperbolic map
- hyperbolic metric
- hyperbolic motion
- hyperbolic navigation system
- hyperbolic octahedron
- hyperbolic-orthogonal
- hyperbolic orbit
- hyperbolic paraboloid
- hyperbolic partial differential equation
- hyperbolic plane
- hyperbolic point
- hyperbolic polar sine
- hyperbolic polyhedron
- hyperbolic polynomial
- hyperbolic quaternion
- hyperbolic rotation
- hyperbolic secant
- hyperbolic secant distribution
- hyperbolic sector
- hyperbolic set
- hyperbolic sine
- hyperbolic sine integral
- hyperbolic small dodecahedral honeycomb
- hyperbolic space
- hyperbolic tangent
- hyperbolic tetrahedron
- hyperbolic time chamber
- hyperbolic trajectory
- hyperbolic triangle
- hyperbolic umbilic catastrophe
- hyperboloid
- quasihyperbolic

#### Translations

pertaining to a hyperbola

- Czech: nadsazený , nadnesený
- French: hyperbolique
- German: übertrieben
- Swedish: hyperbolisk

pertaining to a mathematical hyperbola

- Czech: hyperbolický
- French: hyperbolique
- German: hyperbolisch
- Polish: hiperboliczny
- Spanish: hiperbólico

# Extensive Definition

distinguish hyperbole

In mathematics, a hyperbola
(Greek ,
"over-thrown") is a type of conic
section defined as the intersection between a right circular
conical
surface and a plane
which cuts through both halves of the cone.

It may also be defined as the locus of
points where the difference in the distance to two fixed points
(called the foci) is
constant. That fixed difference in distance is two times a where a
is the distance from the center of the hyperbola to the vertex of
the nearest branch of the hyperbola. a is also known as the
semi-major axis of the hyperbola. The foci lie on the transverse
axis and their midpoint is called the center.

For a simple geometric proof that the two
characterizations above are equivalent to each other, see Dandelin
spheres.

Algebraically, a hyperbola is a curve in the
Cartesian
plane defined by an equation of the form

- A x^2 + B xy + C y^2 + D x + E y + F = 0

The graph of two variables varying inversely on
the Cartesian coordinate plane is a hyperbola.

## Definitions

The first two were listed above:- The intersection between a right circular conical surface and a plane which cuts through both halves of the cone.

- The locus of points where the difference in the distance to two fixed points (called the foci) is constant.

- 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.

A hyperbola comprises two disconnected curves called its arms or branches
which separate the foci. At large distances from the foci the
hyperbola begins to approximate two lines, known as asymptotes. The asymptotes
cross at the center of the hyperbola and have slope \pm \frac for
an East-West opening hyperbola or \pm \frac for a North-South
opening hyperbola.

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.
Also, if rays are directed towards one of the foci from the
exterior of the hyperbola, they will be reflected towards the other
focus.

The simplest example of these are the hyperbolas

- y=\frac\,.

### Polar

East-west opening hyperbola:- r^2 =a\sec 2\theta \,

- r^2 =-a\sec 2\theta \,

- r^2 =a\csc 2\theta \,

- r^2 =-a\csc 2\theta \,

In all formulas the center is at the pole, and a
is the semi-major axis and semi-minor axis.

### Parametric

East-west opening hyperbola:- \begin

North-south opening hyperbola:

- \begin

In all formulae (h,k) are the center coordinates
of the hyperbola, a is the length of the semi-major axis, and b is
the length of the semi-minor axis.

## See also

## External links

hyperbolic in Afrikaans: Hiperbool

hyperbolic in Arabic: قطع زائد

hyperbolic in Bulgarian: Хипербола

hyperbolic in Catalan: Hipèrbola

hyperbolic in Czech: Hyperbola

hyperbolic in Danish: Hyperbel

hyperbolic in German: Hyperbel
(Mathematik)

hyperbolic in Estonian: Hüperbool

hyperbolic in Modern Greek (1453-): Υπερβολή
(γεωμετρία)

hyperbolic in Spanish: Hipérbola

hyperbolic in Esperanto: Hiperbolo

hyperbolic in Persian: هذلولی

hyperbolic in French: Hyperbole
(mathématiques)

hyperbolic in Korean: 쌍곡선

hyperbolic in Hindi: अति परवलय

hyperbolic in Indonesian: Hiperbola
(matematika)

hyperbolic in Italian: Iperbole
(geometria)

hyperbolic in Hebrew: היפרבולה

hyperbolic in Georgian: ჰიპერბოლა

hyperbolic in Lithuanian: Hiperbolė

hyperbolic in Hungarian: Hiperbola

hyperbolic in Dutch: Hyperbool (wiskunde)

hyperbolic in Japanese: 双曲線

hyperbolic in Norwegian: Hyperbel

hyperbolic in Polish: Hiperbola
(matematyka)

hyperbolic in Portuguese: Hipérbole

hyperbolic in Romanian: Hiperbolă

hyperbolic in Russian: Гипербола
(математика)

hyperbolic in Slovak: Hyperbola

hyperbolic in Slovenian: Hiperbola

hyperbolic in Serbian: Хипербола

hyperbolic in Finnish: Hyperbeli

hyperbolic in Swedish: Hyperbel

hyperbolic in Tamil: அதிபரவளைவு

hyperbolic in Vietnamese: Hyperbol

hyperbolic in Chinese: 双曲线