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

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.

The hyperbola has mirror symmetry about its principal axes, and is also symmetric under a 180 ° turn about its center.

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

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.

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.

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.

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.

If, the angle 2θ between the asymptotes equals 90 ° and the hyperbola is said to be rectangular or equilateral.

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

* A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolas having those lines as common asymptotes.

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

The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:

The direction of the asymptotic direction are the same as the asymptotes of the hyperbola of the Dupin indicatrix.

: made the quadrature of a hyperbola to its asymptotes, and showed that as the area increased in arithmetic series the abscissas increased in geometric series.

* Polar distance ( astronomy ), an astronomical term associated with the celestial equatorial coordinate system Σ ( α, δ ) ellipse and lower, a hyperbola

# a lemma upon the Surface Loci of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a conic, and is followed by proofs that the conic is a parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than 1 ( the first recorded proofs of the properties, which do not appear in Apollonius ).

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:

A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If e is between zero and one the conic is an ellipse ; if e = 1 the conic is a parabola ; and if e > 1 the conic is a hyperbola.

To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distance between the center of this circle and the focus ; thus, the focus is outside the directrix circle.

If the polar distance of the Sun is equal to the observer's latitude, the shadow path of a gnomon's tip on a sundial will be a parabola ; at higher latitudes it will be an ellipse and lower, a hyperbola.

Receivers identified which hyperbola they were on and a position could be plotted at the intersection of the hyperbola from different patterns, usually by using the pair with the angle of cut closest to orthogonal as possible.

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

The word " hyperbola " derives from the Greek, meaning " over-thrown " or " excessive ", from which the English term hyperbole also derives.

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.

Rutherford realized this, and also realized that actual impact of the alphas on gold causing any force-deviation from that of the 1 / r coulomb potential would change the form of his scattering curve at high scattering angles ( the smallest impact parameters ) from a hyperbola to something else.

* 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 property of the radius being orthogonal to the tangent at the curve, is extended from the circle to the hyperbola by the hyperbolic orthogonal concept.

Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonal to a ray from the origin traces out a hyperbola.

The quadrature of the hyperbola is the evaluation of the area swept out by a radial segment from the origin as the terminus moves along the hyperbola, just the topic of hyperbolic angle.

The excess velocity vector for a hyperbola is displaced from the periapsis tangent by a characteristic angle, therefore the periapsis injection burn must lead the planetary departure point by the same angle:

In reflection seismology, stacking velocity is the value of velocity obtained from the best-fit hyperbola analysis.

* Trilinear coordinates for the center of the Kiepert hyperbola are ( b < sup > 2 </ sup > − c < sup > 2 </ sup >)< sup > 2 </ sup >/ a: ( c < sup > 2 </ sup > − a < sup > 2 </ sup >)< sup > 2 </ sup >/ b: ( a < sup > 2 </ sup > − b < sup > 2 </ sup >)< sup > 2 </ sup >/ c

* Trilinear coordinates for the center of the Jeřábek hyperbola are cos A sin < sup > 2 </ sup >( B − C ): cos B sin < sup > 2 </ sup >( C − A ): cos C sin < sup > 2 </ sup >( A − B )

The nine-point center is indexed as X ( 5 ), the Feuerbach point, as X ( 11 ), the center of the Kiepert hyperbola as X ( 115 ), and the center of the Jeřábek hyperbola as X ( 125 ).

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.

Note also that the one point on the nine-point circle that is the center of this rectangular hyperbola will have four different definitions dependent on which of the four possible triangles is used as the reference triangle.

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.