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This article is not about hyperbole, which see.
A hyperbola is a type of conic section.
- Geometrically, it is defined as the intersection between a cone and a plane which cuts through both halves of the cone.
- Analytically, 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.
For a simple geometric proof that the two charaterizations above are equivalent to each other, see Dandelin spheres.
- It can also be defined as 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. These foci lie on the transverse axis and their midpoint
is called the center.
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 equilateral or 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 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.
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):
(center (h, k) )
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Equations (polar):
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Equations (parametric):
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See also
- Ellipse, parabola, conic section, Dandelin
spheres, hyperbole
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