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

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.

is a hyperbola for every nonzero a in R. The hyperbola consists of a right and left branch passing through ( a, 0 ) and (− a, 0 ).

In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections ( i. e. either a line, a circle or ellipse, a parabola or a hyperbola ).

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

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.

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.

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

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.

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 ).

It is the form of the fundamental property ( expressed in terms of the " application of areas ") that leads him to give these curves their names: parabola, ellipse, and 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

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

A hyperbola aligned in this way is called an " East-West opening hyperbola ".

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

It was formerly also called hyperbolic logarithm, as it corresponds to the area under a hyperbola.

In projective geometry, Pascal's theorem ( aka Hexagrammum Mysticum Theorem ) states that if an arbitrary six points are chosen on a conic ( i. e., ellipse, parabola or hyperbola ) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon ( extended if necessary ) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.

The case a = 1 is called the unit 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:

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.

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.

Once the required excess velocity v < sub >∞</ sub > ( sometimes called characteristic velocity ) is determined, the injection velocity at periapsis for a hyperbola is:

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.

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 ).

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

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.

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