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Encyclopedia > Hyperbola


In mathematics, a hyperbola (Greek ὑπερβολή literally 'overshooting' or 'excess') 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...


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. In mathematics, a locus (plural loci) is a collection of points which share a common property. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... In geometry, the focus (pl. ...


For a simple geometric proof that the two characterizations above are equivalent to each other, see Dandelin spheres. Dandelin Spheres—graphics by Hop David In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus: Each Dandelin sphere touches, but does not cross, both the plane and the cone. ...


Algebraically, a hyperbola is a curve in the Cartesian plane defined by an equation of the form Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

such that B2 > 4AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the hyperbola, exists.

Contents

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{b}{a} for an East-West opening hyperbola or pm frac{a}{b} for a North-South opening hyperbola. In mathematics, a locus (plural loci) is a collection of points which share a common property. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... (This page refers to eccentricity in mathematics. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... An asymptote is a straight line or curve which a curve approaches as one moves along the curve. ...


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 Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----*---> A B C In geometric... In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ...

Conjugate unit rectangular hyperbolas
Conjugate unit rectangular hyperbolas

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. The rectangular hyperbola x2-y2 and its conjugate, having the same asymptotes. ... The rectangular hyperbola x2-y2 and its conjugate, having the same asymptotes. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...


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, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ... For other uses, see Ellipse (disambiguation). ... 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

North-south opening hyperbola centered at (h,k):

frac{left( y-k right)^2}{a^2} - frac{left( x-h 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. The semi-major axis of an ellipse In geometry, the term 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) applies to ellipses and hyperbolas. ...


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 (This page refers to eccentricity in mathematics. ...

e = sqrt{1+frac{b^2}{a^2}}

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

left(hpm c, kright) where c is given by c2 = a2 + b2

and for a north-south opening hyperbola are given by

left( h, kpm cright) again with c2 = a2 + b2

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

(x-h)(y-k) = c ,
A graph of the rectangular hyperbola, .
A graph of the rectangular hyperbola, y=tfrac{1}{x}.

The simplest example of these are the hyperbolas Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

y=frac{m}{x},.

Polar

East-west opening hyperbola:

r^2 =asec 2theta ,

North-south opening hyperbola:

r^2 =-asec 2theta ,

Northeast-southwest opening hyperbola:

r^2 =acsc 2theta ,

Northwest-southeast opening hyperbola:

r^2 =-acsc 2theta ,

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


Parametric

East-west opening hyperbola:

begin{matrix} x = asec t + h  y = btan t + k  end{matrix} qquad mathrm{or} qquadbegin{matrix} x = pm acosh t + h  y = bsinh t + k  end{matrix}

North-south opening hyperbola:

begin{matrix} x = atan t + h  y = bsec t + k  end{matrix} qquad mathrm{or} qquadbegin{matrix} x = asinh t + h  y = pm bcosh t + k  end{matrix}

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


See also

For other uses, see Ellipse (disambiguation). ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... Circle illustration This article is about the shape and mathematical concept of circle. ... Dandelin Spheres—graphics by Hop David In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus: Each Dandelin sphere touches, but does not cross, both the plane and the cone. ... 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 (b,1/b) and by the hyperbola xy = 1. ... 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 an orbit with the eccentricity greater than 1. ... In mathematics, a subset of a manifold is said to have hyperbolic structure when its tangent bundle may be split into two invariant subbundles, one of which is contracting, and the other expanding. ... Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation  (hyperboloid of one sheet), or  (hyperboloid of two sheets) If, and only if, a = b, it is a hyperboloid of revolution. ... Multilateration, also known as hyperbolic positioning, is the process of locating an object by accurately computing the time difference of arrival (TDOA) of a signal emitted from the object to three or more receivers. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...

External links


  Results from FactBites:
 
Hyperbola (478 words)
Hyperbola is commonly defined as the locus of points P such that the difference of the distances from P to two fixed points F1, F2 (called foci) are constant.
The vertexes are the intersections of the hyperbola and its axis.
A rectangular hyperbola is a hyperbola with eccentricity Sqrt[2]≈1.4142.
Hyperbola - LoveToKnow 1911 (743 words)
The hyperbola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate hyperbola.
In the rectangular hyperbola a =b; hence its equation is x 2 - y 2 = O. The equations to the asymptotes are = t y/b and x = =y respectively.
Referred to the asymptotes as axes the general equation becomes xy 2 obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola.
  More results at FactBites »

 
 

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