Hyperbola

Not to be confused with hyperbole. Hyperbole (haɪˈpɝːbəli hye-PER-buh-lee; "HYE-per-bowl" is a mispronunciation comes from Greek "υπερβολή" (meaning exaggeration and is a

Hyperbola in terms of its conic section

In mathematics, a hyperbola (Greek ὑπερβολή, "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. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Geometry, a ( general) conical surface is the unbounded Surface formed by the union of all the straight lines that pass through a fixed

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. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property Distance is a numerical description of how far apart objects are In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle 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. In Geometry, a nondegenerate Conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus Each

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

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

such that $B 2 > 4 A C$ , where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the hyperbola, exists. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

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

1 Definitions
2 Equations
3 See also
4 External links

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. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property

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. A ratio is an expression which compares quantities relative to each other This constant is the eccentricity of the hyperbola. In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section.

A hyperbola comprises two disconnected curves called its arms or branches which separate the foci. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object At large distances from the foci the hyperbola begins to approximate two lines, known as asymptotes. An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity The asymptotes cross at the center of the hyperbola and have slope $\pm \frac{b}{a}$ for an East-West opening hyperbola or $\pm \frac{a}{b}$ 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. In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. Also, if rays are directed towards one of the foci from the exterior of the hyperbola, they will be reflected towards the other focus.

Conjugate unit rectangular hyperbolas (blue and green) and asymptotes (red)

A special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angles. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy= $c 2$ , where c is a constant (see figure below). The point on the curve nearest the origin is $(\sqrt c, \sqrt c )$ . Also, a line passing through the origin and this point is perpendicular to the tangent line.

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. In Mathematics, parametric equations are a method of defining a curve In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions

If on the hyperbola equation one switches x and y, the conjugate hyperbola is obtained. A hyperbola and its conjugate have the same asymptotes.

Equations

Cartesian

East-west opening hyperbola centered at (h,k):

$\frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1$

The major axis runs through the center of the hyperbola and intersects both arms of the hyperbola at the vertices (bend points) of the arms. The foci lie on the extension of the major axis of the hyperbola.

The minor axis runs through the center of the hyperbola and is perpendicular to the major axis.

In both formulas a is the semi-major axis (half the distance between the two arms of the hyperbola measured along the major axis), and b is the semi-minor axis. In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae In Geometry, the semi-minor axis (also semiminor axis) is a Line segment associated with most Conic sections (that is with ellipses and

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the length of the sides tangent to the hyperbola are 2b in length while the sides that run parallel to the line between the foci (the major axis) are 2a in length. Note that b may be larger than a.

If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always 2a.

The eccentricity is given by

$\varepsilon = \sqrt{1+\frac{b^2}{a^2}} = \sec\left(\arctan\left(\frac{b}{a}\right)\right) = \cosh\left(\operatorname{arcsinh}\left(\frac{b}{a}\right)\right)$

If c equals the distance from the center to either focus, then

$\varepsilon = \frac{c}{a}$

where

$c = \sqrt{a^2 + b^2}$ . In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section.

The distance c is known as the linear eccentricity of the hyperbola. The distance between the foci is 2c or 2aε.

The foci for an east-west opening hyperbola are given by

$\left(h\pm c, k\right)$

and for a north-south opening hyperbola are given by

$\left( h, k\pm c\right)$ .

The directrixes for an east-west opening hyperbola are given by

$x = h\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)$

and for a north-south opening hyperbola are given by

$y = k\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)$ .

For rectangular hyperbolas with the coordinate axes parallel to their asymptotes:

$(x-h)(y-k) = c \,$

A graph of the rectangular hyperbola $y=\tfrac{1}{x}$ .

The simplest example of these are the hyperbolas

$y=\frac{m}{x}\,$ .

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:

$\begin{matrix} x = a\sec t + h \\ y = b\tan t + k \\ \end{matrix} \qquad \mathrm{or} \qquad\begin{matrix} x = \pm a\cosh t + h \\ y = b\sinh t + k \\ \end{matrix}$

North-south opening hyperbola:

$\begin{matrix} x = a\tan t + h \\ y = b\sec t + k \\ \end{matrix} \qquad \mathrm{or} \qquad\begin{matrix} x = a\sinh t + h \\ y = \pm b\cosh t + k \\ \end{matrix}$

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.

External links

In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a A hyperbolic sector is a region of the Cartesian plane {( x, y)} bounded by rays from the origin to two points ( a, 1/ a) and ( A hyperbolic angle in standard position is the Angle at (0 0 between the ray to (1 1 and the ray to ( x, 1/ x) where x > 1 In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions In Astrodynamics or Celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1 In Mathematics, a subset of a Manifold is said to have hyperbolic structure with rspect to a map f, when its Tangent bundle may be split into In Mathematics, a hyperboloid is a Quadric, a type of surface in three Dimensions described by the equation {x^2 \over a^2} + Multilateration, also known as hyperbolic positioning, is the process of locating an object by accurately computing the time difference of arrival ( TDOA In Mathematics, a hyperbolic partial differential equation is usually a second-order Partial differential equation (PDE of the form A u_{xx} When a quantity grows towards a Singularity under a finite variation it is said to undergo hyperbolic growth. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(geometry) 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.

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