In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set.

There are therefore two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch.

If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

These solutions yield good rational approximations of the form x / y to the square root of n. In Cartesian coordinates, the equation has the form of a hyperbola ; it can be seen that solutions occur where the curve has integral ( x, y ) coordinates.

A hyperboloid of revolution of one sheet can be obtained by revolving a hyperbola around its semi-minor axis.

A hyperboloid of revolution of two sheets can be obtained by revolving a hyperbola around its semi-major axis.

Any conic that passes through the four orthocentric points can only be a rectangular hyperbola.

This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola ( bisector of its two foci ).

In a hyperbola, a conjugate axis or minor axis of length 2b, corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices ( turning points ) of the hyperbola, with the two axes intersecting at the center of the hyperbola.

Note that in a hyperbola b can be larger than a.

For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola.

A hyperbola can be defined as the locus of points for each of which the absolute value of the difference between the distances to two given foci is a constant.

The same construction can also be applied to the hyperbola.

It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram.

A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them ; however, a degenerate hyperbola consists only of its asymptotes.

As an example of the first failure, reducibility, is not degenerate ( it defines a hyperbola ), but is degenerate because it is reducible – it factors as, and corresponds to two intersecting lines or an " X ".

Note that over the complex numbers factors as and is degenerate because reducible, while defines a non-degenerate curve, an ellipse / hyperbola ( these are not distinct over the complex numbers, because there is no sense of positive or negative ).

For example, the pencil of curves ( 1-dimensional linear system of conics ) defined by is non-degenerate for but is degenerate for concretely, it is an ellipse for two parallel lines for and a hyperbola with < math > a < 0 </ math > – throughout, one axis has length 2 and the other has length which is infinity for

As the foci of a hyperbola merge into one another, the hyperbola becomes a pair of straight lines.

The Agena-Mariner separation injected the Mariner 2 spacecraft into a geocentric escape hyperbola at 26 minutes 3 seconds after lift-off.

He then supposed this cylindrical column of water to be divided into two parts, the first, which he called the " cataract ," being an hyperboloid generated by the revolution of an hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice, and the second the remainder of the water in the cylindrical vessel.

A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows.

Each branch of the hyperbola consists of two arms which become straighter ( lower curvature ) further out from the center of the hyperbola.

For comparison, the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for " deficient " and " comparable "; these terms may refer to the eccentricity of these curves, which is greater than one ( hyperbola ), less than one ( ellipse ) and exactly one ( parabola ), respectively.

The asymptotes of the hyperbola ( red curves ) are shown as blue dashed lines and intersect at the center of the hyperbola, C. The two focal points are labeled F < sub > 1 </ sub > and F < sub > 2 </ sub >, and the thin black line joining them is the transverse axis.

A hyperbola consists of two disconnected curves called its arms or branches.

Outside of the transverse axis but on the same line are the two focal points ( foci ) of the hyperbola.

The line through these five points is one of the two principal axes of the hyperbola, the other being the perpendicular bisector of the transverse axis.

At large distances from the center, the hyperbola approaches two lines, its asymptotes, which intersect at the hyperbola's center.

However, as we have seen, reducing a tachyon's energy increases its speed, so that the single hyperbola formed is of two oppositely charged tachyons with opposite momenta ( same magnitude, opposite sign ) which annihilate each other when they simultaneously reach infinite speed at the same place in space.

* In conic sections, it is said of two ellipses, two hyperbolas, or an ellipse and a hyperbola which share both foci with each other.

* Hyperbolic sector, is a region of the Cartesian plane bounded by rays from the origin to two points ( a, 1 / a ) and ( b, 1 / b ) and by the hyperbola xy

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

The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates ; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray and the constant of inverse proportionality that locates a point on a hyperbola.

A hyperbola meets it at two real points corresponding to the two directions of the asymptotes.

A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes.

So the cissoid of two non-parallel lines is a hyperbola containing the pole.

A similar derivation show that, conversely, any hyperbola is the cissoid of two non-parallel lines relative to any point on it.

In plane geometry, two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola.

Thus, for a given hyperbola and asymptote A, a pair of lines ( a, b ) are hyperbolic orthogonal if there is a pair ( c, d ) such that, and c is the reflection of d across A.

The arms of the hyperbola approach asymptotic lines and the ' right-hand ' arm of one branch of a hyperbola meets the ' left-hand ' arm of the other branch of a hyperbola at the point at infinity ; this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity.