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# Hyperbola

A hyperbola is a type of conic section. It is defined as the set of all points for which the difference in the distance to two fixed points (called the foci) is constant. It can also be defined as the locus of points whose ratio of the distance from one focus to a line (called the directrix) is a constant larger than one. This constant is the eccentricity of the hyperbola.

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.

A special case of the hyperbola is the rectangular hyperbola, in which the asymptotes intersect at right angles. The rectangular hyperbola with the co-ordinate axes as its asymptotes is given by the equation xy=c, where c is a constant.

Just as the equation of the ellipse in parametric form is related to the sine and cosine functions, so the equation of the hyperbola in parametric form is related to the hyperbolic sine and hyperbolic cosine.

A body that has sufficient energy to escape the gravitational field of a massive body moves in a hyperbolic trajectory with the massive body at one of the foci.

Equations (Cartesian):

$\left(\left(x-a\right)/c\right)^2 - \left(\left(y-b\right)/d\right)^2 = 1$
$\left(\left(x-a\right)/c\right)^2 - \left(\left(y-b\right)/d\right)^2 = -1$
$y-a = c/\left(x-b\right)$
$y-a = -c/\left(x-b\right)$

Equations (Polar):

$r^2 = a\,\sec 2t$
$r^2 = -a\,\sec 2t$
$r^2 = a\,\csc 2t$
$r^2 = -a\,\csc 2t$

Equations (Parametric):

$x = a\,\cosh \theta$
$y = b\,\sinh \theta$