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.

The foci of the hyperbola are at the transmitting stations, A and B.

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.

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.

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.

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

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.

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.

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

* 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.

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

There are clear advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola lies across the line at infinity ; and that a parabola is distinguished only by being tangent to the same line.

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 ).

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.

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.

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.

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.

* 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 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 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.

If u < sub > 1 </ sub > is zero or a negative real number, the orbit is a parabola or a hyperbola, respectively.

For points on the hyperbola below the-axis, the area is considered negative ( see: Image: HyperbolicAnimation. gif | animated version with comparison with the trigonometric ( circular ) functions ).

Just as the points ( cos t, sin t ) form a circle with a unit radius, the points ( cosh t, sinh t ) form the right half of the equilateral hyperbola.

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

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.

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 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.

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.

y / r, then uv = xy and the points of the image of the squeeze mapping are on the same hyperbola as ( x, y ) is.

For example, the isogonal conjugate of a line is a circumconic ; specifically, an ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points.

Newton points out here, that if the speed is high enough, the orbit is no longer an ellipse, but is instead a parabola or hyperbola.

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