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Hyperbola

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For hyperbole, the figure of speech, see hyperbole.

In mathematics, a hyperbola is a type of conic section (Greek ὑπερβολή literally 'overshooting' or 'excess') 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.

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.

Contents

Definitions

A hyperbola comprises two disconnected curves called its arms which separate the foci. At large distances from the foci the hyperbola begins to approximate two lines, known as asymptotes.

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.

An ambigenal hyperbola is one of the triple hyperbolas of the second order, having one of its infinite legs falling within an angle formed by the asymptotes, and the other without. 1828 Webster's Dictionary, public domain.

Conjugate unit rectangular hyperbolas
Enlarge
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.

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.

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:
[\frac - \frac = 1]
North-south opening hyperbola:
[\frac - \frac = 1]
In both formulas (h,k) is the center of the hyperbola, a is the semi-major axis (half the distance between the two branches), and b is the semi-minor axis. Note that b may be larger than a.

The eccentricity is given by

[e = \sqrt}]
The foci for an east-west opening hyperbola are given by
[\left(\pm c, 0\right)]
and for a north-south opening hyperbola are given by
[\left( 0, \pm c\right)] where c is given by [c^2 = a^2 + b^2]
For rectangular hyperbolas with the coordinate axes parallel to their asymptotes:
[(x-h)(y-k) = c \,]

Polar

East-west opening hyperbola:
[r^2 =a\sec 2t \,]
North-south opening hyperbola:
[r^2 =-a\sec 2t \,]
Northeast-southwest opening hyperbola:
[r^2 =a\csc 2t \,]
Northwest-southeast opening hyperbola:
[r^2 =-a\csc 2t \,]
In all formulas the center is at the pole, and a is the semi-major and semi-minor axis.

Parametric

East-west opening hyperbola:
[x = a\sec \theta + h\,]
[y = b\tan \theta + k\,]
North-south opening hyperbola:
[x = a\tan \theta + h\,]
[y = b\sec \theta + k\,]
In both formulas (h,k) is the center of the hyperbola, a is the semi-major axis, and b is the semi-minor axis.

See also

References

External links

 

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

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