hyperbola n : an open curve formed by a plane that cuts the base of a right circular cone

English

Etymology

From (huperbolē)

Noun

hyperbola (plural hyperbolas or hyperbolae)

1. A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.

Translations

See also

• circle
• conic section
• ellipse
• parabola

Czech

Noun

1. hyperbola

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

such that B^2 > 4 AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the hyperbola, exists.

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 \,

North-south opening hyperbola:

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

Northeast-southwest opening hyperbola:

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

Northwest-southeast opening hyperbola:

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

x = a\sec t + h \\ y = b\tan t + k \\ \end \qquad \mathrm \qquad\begin x = \pm a\cosh t + h \\ y = b\sinh t + k \\ \end

North-south opening hyperbola:

\begin

x = a\tan t + h \\ y = b\sec t + k \\ \end \qquad \mathrm \qquad\begin x = a\sinh t + h \\ y = \pm b\cosh t + k \\ \end

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

• Parabola
• Circle
• Ellipse
• Hyperbolic sector
• Hyperbolic angle
• Hyperbolic function
• Hyperbolic trajectory
• Hyperbolic structure
• Hyperboloid
• Multilateration
• Hyperbolic partial differential equation
• Hyperbolic growth

External links

• Apollonius' Derivation of the Hyperbola at Convergence
• Mathworld - Hyperbola

arc, bow, catacaustic, catenary, caustic, circle, conchoid, crook, curl, curve, diacaustic, ellipse, festoon, hook, lituus, parabola, sinus, tracery