alt = The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half ; the boundary curve of the slice on the cone is the hyperbola.

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

Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a " North-South opening hyperbola " and has equation

An open orbit has the shape of a hyperbola ( when the velocity is greater than the escape velocity ), or a parabola ( when the velocity is exactly the escape velocity ).

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

When defined over the positive real numbers, has infinitely many minimal elements of the form, one for each positive number ; this set of points forms one of the branches of a hyperbola.

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

A hyperbola has two Dandelin spheres, touching opposite nappes of the cone.

In the study of spacetime the use of the unit hyperbola to calibrate spacio-temporal measurements has become common since Hermann Minkowski discussed it in 1908.

For all real values of the hyperbolic angle θ the split-complex number λ = exp ( jθ ) has norm 1 and lies on the right branch of the unit hyperbola.

) As the hyperbola xy = 1, associated with the hyperbolic angle, has shortest diameter between (− 1, − 1 ) and ( 1, 1 ), it too has semidiameter √ 2.

Erdős ' bound has been improved subsequently: show that, when n / 2 is prime, one can obtain a solution with 3 ( n-2 )/ 2 points by placing points on the hyperbola xy ≡ k ( mod n / 2 ) for a suitable k. Again, for arbitrary n one can perform this construction for a prime near n / 2 to obtain a solution with

More precisely, the reciprocal function has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as is infinity: any similar graph is said to exhibit hyperbolic growth.

This is the general formula for a conic section that has one focus at the origin ; corresponds to a circle, < math > e < 1 </ math > corresponds to an ellipse, corresponds to a parabola, and corresponds to a hyperbola.

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.

Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis of a Cartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±, where b = a × tan ( θ ) and where θ is the angle between the transverse axis and either asymptote.

It is one of the axes of symmetry for the curve: in an ellipse, the shorter one ; in a hyperbola, the one that does not intersect the hyperbola.

Among the earliest recognition of a squeeze symmetry was the 1647 discovery by Grégoire de Saint-Vincent that the area under a hyperbola ( concretely, the curve given by xy = k ) is the same over as over when a / b = c / d – this corresponds to the area under a hyperbola being preserved under hyperbolic rotation, and was a key step in the development of the logarithm.

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.

The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line ( shown in green ).

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

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

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.

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.

A charged tachyon traveling in a vacuum therefore undergoes a constant proper time acceleration and, by necessity, its worldline forms a hyperbola in space-time.

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.

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.

* If the Specific orbital energy is positive, the body's kinetic energy is greater than its potential energy: The orbit is thus open, following a hyperbola with focus at the other body.

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.

So if a rectangular hyperbola is drawn through four orthocentric points it will have one fixed center on the common nine-point circle but it will have four perspectors one on each of the orthic axes of the four possible triangles.

As E. T. Whittaker wrote in 1910, " hyperbola is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters.

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.

Either half of the minor axis is called the semi-minor axis, of length b. Denoting the semi-major axis length ( distance from the center to a vertex ) as a, the semi-minor and semi-major axes ' lengths appear in the equation of the hyperbola relative to these axes as follows:

In the Cartesian plane, reference is sometimes made to a unit circle or a unit hyperbola.

A parabola is an ellipse that is tangent to the line at infinity Ω, and the hyperbola is an ellipse that crosses Ω.

The acceleration at position r is equal to the curvature of the hyperbola at fixed r, and like the curvature of the nested circles in polar coordinates, it is equal to 1 / r.

Each cone of light is drawing a branch of a hyperbola on a nearby vertical wall.

The hyperbola is one of the four kinds of conic section, formed by the intersection of a plane and a cone.

If the angle between the plane and the axis is less than the angle between the line on the cone and the axis, or if the plane is parallel to the axis, then the conic is a hyperbola.

The term hyperbola is believed to have been coined by Apollonius of Perga ( ca.

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.

Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does.

Because of the minus sign in some of the formulas below, it is also called the imaginary axis of the hyperbola.