Hyperbola

Overview

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

a

**hyperbola**is a curve, specifically a smooth

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

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. A hyperbola has two pieces, called connected component

Connected component (graph theory)

In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. For example, the graph shown in the illustration on the right has three connected components...

s or branches, which are mirror images of each other and resembling two infinite bows

Bow (weapon)

The bow and arrow is a projectile weapon system that predates recorded history and is common to most cultures.-Description:A bow is a flexible arc that shoots aerodynamic projectiles by means of elastic energy. Essentially, the bow is a form of spring powered by a string or cord...

. The hyperbola is one of the four kinds of conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

, formed by the intersection of a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

and a cone

Cone (geometry)

A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

.

Unanswered Questions

Encyclopedia

In mathematics

a

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. A hyperbola has two pieces, called connected component

s or branches, which are mirror images of each other and resembling two infinite bows

. The hyperbola is one of the four kinds of conic section

, formed by the intersection of a plane

and a cone

. The other conic sections are the parabola

, the ellipse

, and the circle

(the circle is a special case of the ellipse). Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a line on the surface of the cone makes with the axis of the 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.

Hyperbolas arise in practice in many ways: as the curve representing the function in the Cartesian plane, as the appearance of a circle

viewed from within it, as the path followed by the shadow of the tip of a sundial, as the shape of an open orbit (as distinct from a closed and hence elliptical orbit), such as the orbit of a spacecraft

during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition comet

(one travelling too fast to ever return to the solar system), as the scattering trajectory

of a subatomic particle

(acted on by repulsive instead of attractive forces but the principle is the same), and so on.

Each branch of the hyperbola consists of two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms one from each branch tend in the limit to a common line, called the asymptote

of those two arms. 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. In the case of the curve the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical object

s have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry

(Lobachevsky's celebrated non-Euclidean geometry

), hyperbolic function

s (sinh, cosh, tanh, etc.), and gyrovector space

s (a geometry used in both relativity and quantum mechanics which is not Euclidean).

, meaning "over-thrown" or "excessive", from which the English term hyperbole

also derives. The term hyperbola is believed to have been coined by Apollonius of Perga

(ca. 262 BC

–ca. 190 BC

) in his definitive work on the conic section

s, the

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 is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse

does. A hyperbola consists of two disconnected curve

s called its

The points on the two branches that are closest to each other are called their

. Outside of the transverse axis but on the same line are the two

of the transverse axis. The hyperbola has mirror symmetry

about its principal axes, and is also symmetric under a 180° turn about its center.

At large distances from the center, the hyperbola approaches two lines, its asymptote

s, 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. Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the

, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±, where

A

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

If the transverse axis of any hyperbola is aligned with the

A hyperbola aligned in this way is called an "East-West opening hyperbola". Likewise, a hyperbola with its transverse axis aligned with the

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. For convenience, hyperbolas are usually analyzed in terms of their centered East-West opening form.

The shape of a hyperbola is defined entirely by its eccentricity

ε, which is a dimensionless number always greater than one. The distance

For example, the eccentricity of a rectangular hyperbola , equals the square root

of two: ε = .

Every hyperbola has a

A few other lengths are used to describe hyperbolas. Consider a line perpendicular to the transverse axis (i.e., parallel to the conjugate axis) that passes through one of the hyperbola's foci. The line segment connecting the two intersection points of this line with the hyperbola is known as the

between a right circular conical surface

and a plane that cuts through both halves of the cone. The other major types of conic sections are the ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

and the parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

; in these cases, the plane cuts through only one half of the double cone. If the plane is parallel to the axis of the double cone and passes through its central apex, a degenerate hyperbola

results that is simply two straight lines that cross at the apex point.

is a constant equal to 2

; this is the problem of determining position from the

.

This definition may be expressed also in terms of tangent circles

. The center of any circles externally tangent to two given circles lies on a hyperbola, whose foci are the centers of the given circles and where the vertex distance 2

A proof that this characterization

of the hyperbola is equivalent to the conic-section characterization can be done without coordinate geometry by means of Dandelin spheres

.

of the distances to one focus and to a line

(called the directrix) is a constant that is larger than 1. This constant is the eccentricity

of the hyperbola. By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex.

, respectively. The

The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius

Since the eccentricity of a hyperbola is always greater than one, the center

This definition implies that the hyperbola is both the locus

of the poles of the tangent lines to the circle

of the polar lines of the points on

provided that the constants

A special case of a hyperbola—the

This determinant Δ is sometimes called the discriminant

of the conic section.

The center (

In terms of new coordinates, and , the defining equation of the hyperbola can be written

The principal axes of the hyperbola make an angle Φ with the positive

Rotating the coordinate axes so that the

The major and minor semiaxes

where λ

For comparison, the corresponding equation for a degenerate hyperbola is

The tangent line to a given point (

where

The normal line

to the hyperbola at the same point is given by the equation

The normal line is perpendicular to the tangent line, and both pass through the same point (

From the equation

the basic property that with and being the distances from a point to the left focus and the right focus one has for a point on the right branch that

and for a point on the left branch that

can be proved as follows:

If x,y is a point on the hyperbola the distance to the left focal point is

To the right focal point the distance is

If x,y is a point on the right branch of the hyperbola then and

Subtracting these equations one gets

If x,y is a point on the left branch of the hyperbola then and

Subtracting these equations one gets

the distance from a point on the left branch of the hyperbola to the left focal point is

.

Introducing polar coordinates with origin at the left focal point the coordinates relative the canonical coordinate system are

and the equation above takes the form

from which follows that

This is the representation of the near branch of a hyperbola in polar coordinates with respect to a focal point.

The polar angle of a point on a hyperbola relative the near focal point as described above is called the

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

, a hyperbola can be constructed using a taut thread. A straightedge of length

A second construction uses intersecting circles, but is likewise based on the constant difference of distances to the foci. Consider a hyperbola with two foci

A third construction relies on the definition of the hyperbola as the reciprocation of a circle. Consider the circle centered on the center of the hyperbola and of radius

of the hyperbola.

from the hyperbola, the light-ray appears to have come from the other focus. Equivalently, by reversing the direction of the light, rays directed at one of the foci from the exterior of the hyperbola are reflected towards the other focus. This property is analogous to the property of ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s that a ray emerging from one focus is reflected from the ellipse directly

Tangent lines to a hyperbola have another remarkable geometrical property. If a tangent line at a point

and cosine functions give a parametric equation

for the ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

, so the hyperbolic sine

and hyperbolic cosine

give a parametric equation for the hyperbola.

As

one has for any value of that the point

satisfies the equation

which is the equation of a hyperbola relative its canonical coordinate system.

When

The left branch for which is in the same way obtained as

In the figure the points given by

for

on the left branch of a hyperbola with eccentricity 1.2 are marked as dots.

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s and parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

s. Since the parabola may be seen as a limiting case poised exactly between an ellipse and a hyperbola, there are effectively only two major types, ellipses and hyperbolas. These two types are related in that formulae for one type can often be applied to the other.

The canonical equation for a hyperbola is

Any hyperbola can be rotated so that it is east-west opening and positioned with its center at the origin, so that the equation describing it is this canonical equation.

The canonical equation for the hyperbola may be seen as a version of the corresponding ellipse equation

in which the semi-minor axis

length

Similarly, the parametric equations for a hyperbola and an ellipse are expressed in terms of hyperbolicIn mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...

and trigonometric function

s, respectively, which are again related by an imaginary number, e.g.,

Hence, many formulae for the ellipse can be extended to hyperbolas by adding the imaginary unit

. The

When the circle directly faces the viewer, the viewer's lens is on-axis, meaning on the line normal to the circle through its center (think of the axle of a wheel). The rays of light from the circle through the lens to the image plane then form a cone with circular cross section whose apex is the lens. The image plane concretely realizes the abstract cutting plane in the conic section model.

When in addition the viewer directly faces the circle, the circle is rendered faithfully on the image plane without perspective distortion, namely as a scaled-down circle. When the viewer turns attention or gaze away from the center of the circle the image plane then cuts the cone in an ellipse, parabola, or hyperbola depending on how far the viewer turns, corresponding exactly to what happens when the surface cutting the cone to form a conic section is rotated.

A parabola arises when the lens plane is tangent to (touches) the circle. A viewer with perfect 180-degree wide-angle vision will see the whole parabola; in practice this is impossible and only a finite portion of the parabola is captured on the film or retina.

When the viewer turns further so that the lens plane cuts the circle in two points, the shape on the image plane becomes that of a hyperbola. The viewer still sees only a finite curve, namely a portion of one branch of the hyperbola, and is unable to see the second branch at all, which corresponds to the portion of the circle behind the viewer, more precisely, on the same side of the lens plane as the viewer. In practice the finite extent of the image plane makes it impossible to see any portion of the circle near where it is cut by the lens plane. Further back however one could imagine rays from the portion of the circle well behind the viewer passing through the lens, were the viewer transparent. In this case the rays would pass through the image plane before the lens, yet another impracticality ensuring that no portion of the second branch could possibly be visible.

The tangents to the circle where it is cut by the lens plane constitute the asymptotes of the hyperbola. Were these tangents to be drawn in ink in the plane of the circle, the eye would perceive them as asymptotes to the visible branch. Whether they converge in front of or behind the viewer depends on whether the lens plane is in front of or behind the center of the circle respectively.

If the circle is drawn on the ground and the viewer gradually transfers gaze from straight down at the circle up towards the horizon, the lens plane eventually cuts the circle producing first a parabola then a hyperbola on the image plane. As the gaze continues to rise the asymptotes of the hyperbola, if realized concretely, appear coming in from left and right, swinging towards each other and converging at the horizon when the gaze is horizontal. Further elevation of the gaze into the sky then brings the point of convergence of the asymptotes towards the viewer.

By the same principle with which the back of the circle appears on the image plane were all the physical obstacles to its projection to be overcome, the portion of the two tangents behind the viewer appear on the image plane as an extension of the visible portion of the tangents in front of the viewer. Like the second branch this extension materializes in the sky rather than on the ground, with the horizon marking the boundary between the physically visible (scene in front) and invisible (scene behind), and the visible and invisible parts of the tangents combining in a single X shape. As the gaze is raised and lowered about the horizon, the X shape moves oppositely, lowering as the gaze is raised and vice versa but always with the visible portion being on the ground and stopping at the horizon, with the center of the X being on the horizon when the gaze is horizontal.

All of the above was for the case when the circle faces the viewer, with only the viewer's gaze varying. When the circle starts to face away from the viewer the viewer's lens is no longer on-axis. In this case the cross section of the cone is no longer a circle but an ellipse (never a parabola or hyperbola). However the principle of conic sections does not depend on the cross section of the cone being circular, and applies without modification to the case of eccentric cones.

It is not difficult to see that even in the off-axis case a circle can appear circular, namely when the image plane (and hence lens plane) is parallel to the plane of the circle. That is, to see a circle as a circle when viewing it obliquely, look not at the circle itself but at the plane in which it lies. From this it can be seen that when viewing a plane filled with many circles, all of them will appear circular simultaneously when the plane is looked at directly.

A common misperception about the hyperbola is that it is a mathematical curve rarely if ever encountered in daily life. The reality is that one sees a hyperbola whenever catching sight of portion of a circle cut by one's lens plane (and a parabola when the lens plane is tangent to, i.e. just touches, the circle). The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas such as

s of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the lemniscate of Bernoulli

; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin. If the center of inversion is chosen at a focus or a vertex of the hyperbola, the resulting inverse curves are a limaçon

or a strophoid

, respectively.

The major axis runs through the center of the hyperbola and intersects both arms of the hyperbola at the vertices (bend points) of the arms. The foci lie on the extension of the major axis of the hyperbola.

The minor axis runs through the center of the hyperbola and is perpendicular to the major axis.

In both formulas

(half the distance between the two arms of the hyperbola measured along the major axis), and

(half the distance between the asymptotes along a line tangent to the hyperbola at a vertex).

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the sides tangent to the hyperbola are

If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always

The eccentricityIn mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

is given by

If

where.

The distance

The foci for an east-west opening hyperbola are given by

and for a north-south opening hyperbola are given by.

The directrices for an east-west opening hyperbola are given by

and for a north-south opening hyperbola are given by.

Relative to this coordinate system one has that

and the range of the true anomaly is:

With polar coordinate relative to the "canonical coordinate system"

one has that

For the right branch of the hyperbola the range of is:

In all formulae (

in two dimensions. These hyperbolas are described by the equation

where the foci are located at a distance

of the Cartesian coordinate system

Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings. For example, the mapping

into two families of orthogonal hyperbolas.

These are equilateral hyperbolas (eccentricity ) with semi-major axis and semi-minor axis given by .

The simplest example of rectangular hyperbolas occurs when the center (

describing quantities

s. On any given day, the sun revolves in a circle on the celestial sphere

, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day. The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a

problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from a LORAN

or GPS transmitters. Conversely, a homing beacon or any transmitter can be located by comparing the arrival times of its signals at two separate receiving stations; such techniques may be used to track objects and people. In particular, the set of possible positions of a point that has a distance difference of 2

is a conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

. In particular, if the total energy

demonstrated the existence of an atomic nucleus

by examining the scattering of alpha particle

s from gold

atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb force

, which satisfies the inverse square law requirement for a Kepler problem.

, a hyperbola can be used to trisect any angle

, an intensely studied problem of geometry. Given an angle, one first draws a circle centered on its middle pointIn mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

ε=2 with that line as its transverse axis and

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

a

**hyperbola**is a curve, specifically a smoothSmooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

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. A hyperbola has two pieces, called connected component

Connected component (graph theory)

In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. For example, the graph shown in the illustration on the right has three connected components...

s or branches, which are mirror images of each other and resembling two infinite bows

Bow (weapon)

The bow and arrow is a projectile weapon system that predates recorded history and is common to most cultures.-Description:A bow is a flexible arc that shoots aerodynamic projectiles by means of elastic energy. Essentially, the bow is a form of spring powered by a string or cord...

. The hyperbola is one of the four kinds of conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

, formed by the intersection of a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

and a cone

Cone (geometry)

A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

. The other conic sections are the parabola

Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

, the ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

, and the circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

(the circle is a special case of the ellipse). Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a line on the surface of the cone makes with the axis of the 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.

Hyperbolas arise in practice in many ways: as the curve representing the function in the Cartesian plane, as the appearance of a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

viewed from within it, as the path followed by the shadow of the tip of a sundial, as the shape of an open orbit (as distinct from a closed and hence elliptical orbit), such as the orbit of a spacecraft

Spacecraft

A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....

during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition comet

Comet

A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

(one travelling too fast to ever return to the solar system), as the scattering trajectory

Rutherford scattering

In physics, Rutherford scattering is a phenomenon that was explained by Ernest Rutherford in 1911, and led to the development of the Rutherford model of the atom, and eventually to the Bohr model. It is now exploited by the materials analytical technique Rutherford backscattering...

of a subatomic particle

Subatomic particle

In physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...

(acted on by repulsive instead of attractive forces but the principle is the same), and so on.

Each branch of the hyperbola consists of two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms one from each branch tend in the limit to a common line, called the asymptote

Asymptote

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

of those two arms. 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. In the case of the curve the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical object

Mathematical object

In mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...

s have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

(Lobachevsky's celebrated non-Euclidean geometry

Non-Euclidean geometry

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

), hyperbolic function

Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...

s (sinh, cosh, tanh, etc.), and gyrovector space

Gyrovector space

A gyrovector space is a mathematical concept for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. This vector-based approach has been developed by Abraham Albert Ungar from the late 1980s onwards...

s (a geometry used in both relativity and quantum mechanics which is not Euclidean).

## History

The word "hyperbola" derives from the GreekGreek language

Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

, meaning "over-thrown" or "excessive", from which the English term hyperbole

Hyperbole

Hyperbole is the use of exaggeration as a rhetorical device or figure of speech. It may be used to evoke strong feelings or to create a strong impression, but is not meant to be taken literally....

also derives. The term hyperbola is believed to have been coined by Apollonius of Perga

Apollonius of Perga

Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

(ca. 262 BC

262 BC

Year 262 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Megellus och Vitulus...

–ca. 190 BC

190 BC

Year 190 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Asiaticus and Laelius...

) in his definitive work on the conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

s, the

*Conics*. For comparison, the other two general conic sections, the ellipseEllipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

and the parabola

Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

, derive from the corresponding Greek words for "deficient" and "comparable"; these terms may refer to the eccentricity

Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

of these curves, which is greater than one (hyperbola), less than one (ellipse) and exactly one (parabola), respectively.

## Nomenclature and features

Similar to a parabolaParabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

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

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

does. A hyperbola consists of two disconnected curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s called its

**arms**or**branches**.The points on the two branches that are closest to each other are called their

*vertices*, and the line segment connecting them is called the*transverse axis*or*major axis*, corresponding to the major diameter of an ellipse. The midpoint of the transverse axis is known as the hyperbola's*center*. The distance*a*from the center to each vertex is called the semi-major axisSemi-major axis

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

. 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 bisectorFocus (geometry)

In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...

of the transverse axis. The hyperbola has mirror symmetry

Reflection symmetry

Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

about its principal axes, and is also symmetric under a 180° turn about its center.

At large distances from the center, the hyperbola approaches two lines, its asymptote

Asymptote

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

s, 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

Degenerate conic

In mathematics, a degenerate conic is a conic that fails to be an irreducible curve...

consists only of its asymptotes. Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the

*x*-axis of a Cartesian coordinate systemCartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

, 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. The distance*b*(not shown) is the length of the perpendicular segment from either vertex to the asymptotes.A

*conjugate axis*of length 2*b*, corresponding to the*minor axis*of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices. Because of the minus sign in some of the formulas below, it is also called the*imaginary axis*of the hyperbola.If , the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be

*rectangular*or*equilateral*. In this special case, the rectangle joining the four points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides*2a**2b*.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 asA 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 equationEvery hyperbola is congruent

Congruence (geometry)

In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...

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

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

(rigidly moved in the plane) so that it is centered at the origin. For convenience, hyperbolas are usually analyzed in terms of their centered East-West opening form.

The shape of a hyperbola is defined entirely by its eccentricity

Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

ε, which is a dimensionless number always greater than one. The distance

*c*from the center to the foci equals*a*ε. The eccentricity can also be defined as the ratio of the distances to either focus and to a corresponding line known as the directrix; hence, the distance from the center to the directrices equals*a*/ε. In terms of the parameters*a*,*b*,*c*and the angle θ, the eccentricity equalsFor example, the eccentricity of a rectangular hyperbola , equals the square root

Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of two: ε = .

Every hyperbola has a

*conjugate hyperbola*, in which the transverse and conjugate axes are exchanged without changing the asymptotes. The equation of the conjugate hyperbola of is . If the graph of the conjugate hyperbola is rotated 90° to restore the east-west opening orientation (so that*x*becomes*y*and vice versa), the equation of the resulting rotated conjugate hyperbola is the same as the equation of the original hyperbola except with*a*and*b*exchanged. For example, the angle θ of the conjugate hyperbola equals 90° minus the angle of the original hyperbola. Thus, the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities unless θ = 45° (a rectangular hyperbola). Hence, the conjugate hyperbola does*not*in general correspond to a 90° rotation of the original hyperbola; the two hyperbolas are generally different in shape.A few other lengths are used to describe hyperbolas. Consider a line perpendicular to the transverse axis (i.e., parallel to the conjugate axis) that passes through one of the hyperbola's foci. The line segment connecting the two intersection points of this line with the hyperbola is known as the

*latus rectum*and has a length . The*semi-latus rectum**l*is half of this length, i.e., . The*focal parameter**p*is the distance from a focus to its corresponding directrix, and equals .### Conic section

A hyperbola may be defined as the curve of intersectionIntersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

between a right circular conical surface

Conical surface

In geometry, a conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex...

and a plane that cuts through both halves of the cone. The other major types of conic sections are the ellipse

Ellipse

and the parabola

Parabola

; in these cases, the plane cuts through only one half of the double cone. If the plane is parallel to the axis of the double cone and passes through its central apex, a degenerate hyperbola

Degenerate conic

In mathematics, a degenerate conic is a conic that fails to be an irreducible curve...

results that is simply two straight lines that cross at the apex point.

### Difference of distances to foci

A hyperbola may be defined equivalently as the locus of points where the absolute value of the*difference*of the distances to the two fociFocus (geometry)

In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

is a constant equal to 2

*a*, the distance between its two vertices. This definition accounts for many of the hyperbola's applications, such as trilaterationTrilateration

In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...

; this is the problem of determining position from the

*difference*in arrival times of synchronized signals, as in GPSGlobal Positioning System

The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...

.

This definition may be expressed also in terms of tangent circles

Tangent circles

In geometry, tangent circles are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external...

. The center of any circles externally tangent to two given circles lies on a hyperbola, whose foci are the centers of the given circles and where the vertex distance 2

*a*equals the difference in radii of the two circles. As a special case, one given circle may be a point located at one focus; since a point may be considered as a circle of zero radius, the other given circle—which is centered on the other focus—must have radius 2*a*. This provides a simple technique for constructing a hyperbola, as shown below. It follows from this definition that a tangent line to the hyperbola at a point**P**bisects the angle formed with the two foci, i.e., the angle**F**_{1}**P****F**_{2}. Consequently, the feet of perpendiculars drawn from each focus to such a tangent line lies on a circle of radius*a*that is centered on the hyperbola's own center.A proof that this characterization

Characterization (mathematics)

In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...

of the hyperbola is equivalent to the conic-section characterization can be done without coordinate geometry by means of Dandelin spheres

Dandelin spheres

In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin...

.

### Directrix and focus

A hyperbola can be defined as the locus of points for which the ratioRatio

In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

of the distances to one focus and to a line

Line (geometry)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

(called the directrix) is a constant that is larger than 1. This constant is the eccentricity

Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

of the hyperbola. By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex.

### Reciprocation of a circle

The reciprocation of a circleCircle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

*B*in a circle*C*always yields a conic section such as a hyperbola. The process of "reciprocation in a circle*C*" consists of replacing every line and point in a geometrical figure with their corresponding pole and polarPole and polar

In geometry, the terms pole and polar are used to describe a point and a line that have a unique reciprocal relationship with respect to a given conic section...

, respectively. The

*pole*of a line is the inversion of its closest point to the circle*C*, whereas the polar of a point is the converse, namely, a line whose closest point to*C*is the inversion of the point.The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius

*r*of reciprocation circle*C*. If**B**and**C**represent the points at the centers of the corresponding circles, thenSince the eccentricity of a hyperbola is always greater than one, the center

**B**must lie outside of the reciprocating circle*C*.This definition implies that the hyperbola is both the locus

Locus (mathematics)

In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

of the poles of the tangent lines to the circle

*B*, as well as the envelopeEnvelope (mathematics)

In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...

of the polar lines of the points on

*B*. Conversely, the circle*B*is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to*B*have no (finite) poles because they pass through the center**C**of the reciprocation circle*C*; the polars of the corresponding tangent points on*B*are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle*B*that are separated by these tangent points.### Quadratic equation

A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates (*x*,*y*) of the planeprovided that the constants

*A*_{xx},*A*_{xy},*A*_{yy},*B*_{x},*B*_{y}, and*C*satisfy the determinant conditionA special case of a hyperbola—the

*degenerate hyperbola*consisting of two intersecting lines—occurs when another determinant is zeroThis determinant Δ is sometimes called the discriminant

Discriminant

In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

of the conic section.

The center (

*x*_{c},*y*_{c}) of the hyperbola may be determined from the formulaeIn terms of new coordinates, and , the defining equation of the hyperbola can be written

The principal axes of the hyperbola make an angle Φ with the positive

*x*-axis that equalsRotating the coordinate axes so that the

*x*-axis is aligned with the transverse axis brings the equation into its**canonical form**The major and minor semiaxes

*a*and*b*are defined by the equationswhere λ

_{1}and λ_{2}are the roots of the quadratic equationQuadratic equation

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

For comparison, the corresponding equation for a degenerate hyperbola is

The tangent line to a given point (

*x*_{0},*y*_{0}) on the hyperbola is defined by the equationwhere

*E*,*F*and*G*are definedThe normal line

Surface normal

A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

to the hyperbola at the same point is given by the equation

The normal line is perpendicular to the tangent line, and both pass through the same point (

*x*_{0},*y*_{0}).From the equation

the basic property that with and being the distances from a point to the left focus and the right focus one has for a point on the right branch that

and for a point on the left branch that

can be proved as follows:

If x,y is a point on the hyperbola the distance to the left focal point is

To the right focal point the distance is

If x,y is a point on the right branch of the hyperbola then and

Subtracting these equations one gets

If x,y is a point on the left branch of the hyperbola then and

Subtracting these equations one gets

## The true anomaly

In the section above it is shown that using the coordinate system in which the equation of the hyperbola takes its**canonical form**the distance from a point on the left branch of the hyperbola to the left focal point is

.

Introducing polar coordinates with origin at the left focal point the coordinates relative the canonical coordinate system are

and the equation above takes the form

from which follows that

This is the representation of the near branch of a hyperbola in polar coordinates with respect to a focal point.

The polar angle of a point on a hyperbola relative the near focal point as described above is called the

**true anomaly**of the point.## Geometrical constructions

Similar to the ellipseEllipse

, a hyperbola can be constructed using a taut thread. A straightedge of length

*S*is attached to one focus**F**_{1}at one of its corners**A**so that it is free to rotate about that focus. A thread of length*L*=*S*- 2*a*is attached between the other focus**F**_{2}and the other corner**B**of the straightedge. A sharp pencil is held up against the straightedge, sandwiching the thread tautly against the straightedge. Let the position of the pencil be denoted as**P**. The total length*L*of the thread equals the sum of the distances*L*_{2}from**F**_{2}to**P**and*L*_{B}from**P**to**B**. Similarly, the total length*S*of the straightedge equals the distance*L*_{1}from**F**_{1}to**P**and*L*_{B}. Therefore, the difference in the distances to the foci, equals the constant 2*a*A second construction uses intersecting circles, but is likewise based on the constant difference of distances to the foci. Consider a hyperbola with two foci

**F**_{1}and**F**_{2}, and two vertices**P**and**Q**; these four points all lie on the transverse axis. Choose a new point**T**also on the transverse axis and to the right of the rightmost vertex**P**; the difference in distances to the two vertices, = 2*a*, since 2*a*is the distance between the vertices. Hence, the two circles centered on the foci**F**_{1}and**F**_{2}of radius QT and PT, respectively, will intersect at two points of the hyperbola.A third construction relies on the definition of the hyperbola as the reciprocation of a circle. Consider the circle centered on the center of the hyperbola and of radius

*a*; this circle is tangent to the hyperbola at its vertices. A line*g*drawn from one focus may intersect this circle in two points**M**and**N**; perpendiculars to*g*drawn through these two points are tangent to the hyperbola. Drawing a set of such tangent lines reveals the envelopeEnvelope (mathematics)

In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...

of the hyperbola.

## Reflections and tangent lines

The ancient Greek geometers recognized a reflection property of hyperbolas. If a ray of light emerges from one focus and is reflectedReflection (mathematics)

In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...

from the hyperbola, the light-ray appears to have come from the other focus. Equivalently, by reversing the direction of the light, rays directed at one of the foci from the exterior of the hyperbola are reflected towards the other focus. This property is analogous to the property of ellipse

Ellipse

s that a ray emerging from one focus is reflected from the ellipse directly

*towards*the other focus (rather than*away*as in the hyperbola). Expressed mathematically, lines drawn from each focus to the same point on the hyperbola intersect it at equal angles; the tangent line to a hyperbola at a point**P**bisects the angle formed with the two foci, F_{1}PF_{2}.Tangent lines to a hyperbola have another remarkable geometrical property. If a tangent line at a point

**T**intersects the asymptotes at two points**K**and**L**, then**T**bisects the line segment KL, and the product of distances to the hyperbola's center, OK×OL is a constant.## Hyperbolic functions and equations

Just as the sineSine

In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

and cosine functions give a parametric equation

Parametric equation

In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

for the ellipse

Ellipse

, so the hyperbolic sine

Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...

and hyperbolic cosine

Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...

give a parametric equation for the hyperbola.

As

one has for any value of that the point

satisfies the equation

which is the equation of a hyperbola relative its canonical coordinate system.

When

*μ*varies over the interval one gets with this formula all points on the right branch of the hyperbola.The left branch for which is in the same way obtained as

In the figure the points given by

for

on the left branch of a hyperbola with eccentricity 1.2 are marked as dots.

## Relation to other conic sections

There are three major types of conic sections: hyperbolas, ellipseEllipse

s and parabola

Parabola

s. Since the parabola may be seen as a limiting case poised exactly between an ellipse and a hyperbola, there are effectively only two major types, ellipses and hyperbolas. These two types are related in that formulae for one type can often be applied to the other.

The canonical equation for a hyperbola is

Any hyperbola can be rotated so that it is east-west opening and positioned with its center at the origin, so that the equation describing it is this canonical equation.

The canonical equation for the hyperbola may be seen as a version of the corresponding ellipse equation

in which the semi-minor axis

Semi-minor axis

In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

length

*b*is imaginary. That is, if in the ellipse equation*b*is replaced by*ib*where*b*is real, one obtains the hyperbola equation.Similarly, the parametric equations for a hyperbola and an ellipse are expressed in terms of hyperbolic

Hyperbolic function

and trigonometric function

Trigonometric function

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s, respectively, which are again related by an imaginary number, e.g.,

Hence, many formulae for the ellipse can be extended to hyperbolas by adding the imaginary unit

*i*in front of the semi-minor axis*b*and the angle. For example, the arclength of a segment of an ellipse can be determined using an incomplete elliptic integral of the second kind. The corresponding arclength of a hyperbola is given by the same function with imaginary parameters*b*and μ, namely,*ib E(iμ, c)*.## Conic section analysis of the hyperbolic appearance of circles

Besides providing a uniform description of circles, ellipses, parabolas, and hyperbolas, conic sections can also be understood as a natural model of the geometry of perspective in the case where the scene being viewed consists of a circle, or more generally an ellipse. The viewer is typically a camera or the human eye. In the simplest case the viewer's lens is just a pinhole; the role of more complex lenses is merely to gather far more light while retaining as far as possible the simple pinhole geometry in which all rays of light from the scene pass through a single point. Once through the lens, the rays then spread out again, in air in the case of a camera, in the vitreous humor in the case of the eye, eventually distributing themselves over the film, imaging device, or retina, all of which come under the heading of image planeImage plane

In 3D computer graphics, the image plane is that plane in the world which is identified with the plane of the monitor. If one makes the analogy of taking a photograph to rendering a 3D image, the surface of the film is the image plane. In this case, the viewing transformation is a projection that...

. The

**lens plane**is a plane parallel to the image plane at the lens; all rays pass through a single point on the lens plane, namely the lens itself.When the circle directly faces the viewer, the viewer's lens is on-axis, meaning on the line normal to the circle through its center (think of the axle of a wheel). The rays of light from the circle through the lens to the image plane then form a cone with circular cross section whose apex is the lens. The image plane concretely realizes the abstract cutting plane in the conic section model.

When in addition the viewer directly faces the circle, the circle is rendered faithfully on the image plane without perspective distortion, namely as a scaled-down circle. When the viewer turns attention or gaze away from the center of the circle the image plane then cuts the cone in an ellipse, parabola, or hyperbola depending on how far the viewer turns, corresponding exactly to what happens when the surface cutting the cone to form a conic section is rotated.

A parabola arises when the lens plane is tangent to (touches) the circle. A viewer with perfect 180-degree wide-angle vision will see the whole parabola; in practice this is impossible and only a finite portion of the parabola is captured on the film or retina.

When the viewer turns further so that the lens plane cuts the circle in two points, the shape on the image plane becomes that of a hyperbola. The viewer still sees only a finite curve, namely a portion of one branch of the hyperbola, and is unable to see the second branch at all, which corresponds to the portion of the circle behind the viewer, more precisely, on the same side of the lens plane as the viewer. In practice the finite extent of the image plane makes it impossible to see any portion of the circle near where it is cut by the lens plane. Further back however one could imagine rays from the portion of the circle well behind the viewer passing through the lens, were the viewer transparent. In this case the rays would pass through the image plane before the lens, yet another impracticality ensuring that no portion of the second branch could possibly be visible.

The tangents to the circle where it is cut by the lens plane constitute the asymptotes of the hyperbola. Were these tangents to be drawn in ink in the plane of the circle, the eye would perceive them as asymptotes to the visible branch. Whether they converge in front of or behind the viewer depends on whether the lens plane is in front of or behind the center of the circle respectively.

If the circle is drawn on the ground and the viewer gradually transfers gaze from straight down at the circle up towards the horizon, the lens plane eventually cuts the circle producing first a parabola then a hyperbola on the image plane. As the gaze continues to rise the asymptotes of the hyperbola, if realized concretely, appear coming in from left and right, swinging towards each other and converging at the horizon when the gaze is horizontal. Further elevation of the gaze into the sky then brings the point of convergence of the asymptotes towards the viewer.

By the same principle with which the back of the circle appears on the image plane were all the physical obstacles to its projection to be overcome, the portion of the two tangents behind the viewer appear on the image plane as an extension of the visible portion of the tangents in front of the viewer. Like the second branch this extension materializes in the sky rather than on the ground, with the horizon marking the boundary between the physically visible (scene in front) and invisible (scene behind), and the visible and invisible parts of the tangents combining in a single X shape. As the gaze is raised and lowered about the horizon, the X shape moves oppositely, lowering as the gaze is raised and vice versa but always with the visible portion being on the ground and stopping at the horizon, with the center of the X being on the horizon when the gaze is horizontal.

All of the above was for the case when the circle faces the viewer, with only the viewer's gaze varying. When the circle starts to face away from the viewer the viewer's lens is no longer on-axis. In this case the cross section of the cone is no longer a circle but an ellipse (never a parabola or hyperbola). However the principle of conic sections does not depend on the cross section of the cone being circular, and applies without modification to the case of eccentric cones.

It is not difficult to see that even in the off-axis case a circle can appear circular, namely when the image plane (and hence lens plane) is parallel to the plane of the circle. That is, to see a circle as a circle when viewing it obliquely, look not at the circle itself but at the plane in which it lies. From this it can be seen that when viewing a plane filled with many circles, all of them will appear circular simultaneously when the plane is looked at directly.

A common misperception about the hyperbola is that it is a mathematical curve rarely if ever encountered in daily life. The reality is that one sees a hyperbola whenever catching sight of portion of a circle cut by one's lens plane (and a parabola when the lens plane is tangent to, i.e. just touches, the circle). The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas such as

*y*= 1/*x*where both branches are on display simultaneously.## Derived curves

Several other curves can be derived from the hyperbola by inversion, the so-called inverse curveInverse curve

In geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·PQ = k2...

s of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the lemniscate of Bernoulli

Lemniscate of Bernoulli

In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is...

; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin. If the center of inversion is chosen at a focus or a vertex of the hyperbola, the resulting inverse curves are a limaçon

Limaçon

In geometry, a limaçon or limacon , also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller...

or a strophoid

Strophoid

In geometry, a strophoid is a curve generated from a given curve C and points A and O as follows: Let L be a variable line passing through O and intersecting C at K. Now let P1 and P2 be the two points on L whose distance from K is the same as the distance from A to K...

, respectively.

### Cartesian coordinates

An east-west opening hyperbola centered at (*h*,*k*) has the equationThe major axis runs through the center of the hyperbola and intersects both arms of the hyperbola at the vertices (bend points) of the arms. The foci lie on the extension of the major axis of the hyperbola.

The minor axis runs through the center of the hyperbola and is perpendicular to the major axis.

In both formulas

*a*is the semi-major axisSemi-major axis

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

(half the distance between the two arms of the hyperbola measured along the major axis), and

*b*is the semi-minor axisSemi-minor axis

In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

(half the distance between the asymptotes along a line tangent to the hyperbola at a vertex).

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the sides tangent to the hyperbola are

*2b*in length while the sides that run parallel to the line between the foci (the major axis) are*2a*in length. Note that*b*may be larger than*a*despite the names*minor*and*major*.If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always

*2a*.The eccentricity

Eccentricity (mathematics)

is given by

If

*c*equals the distance from the center to either focus, thenwhere.

The distance

*c*is known as the**linear eccentricity**of the hyperbola. The distance between the foci is 2*c*or 2*aε*.The foci for an east-west opening hyperbola are given by

and for a north-south opening hyperbola are given by.

The directrices for an east-west opening hyperbola are given by

and for a north-south opening hyperbola are given by.

### Polar coordinates

The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has its origin in a focus and its x-axis pointing towards the origin of the "canonical coordinate system" as illustrated in the figure of the section "True anomaly".Relative to this coordinate system one has that

and the range of the true anomaly is:

With polar coordinate relative to the "canonical coordinate system"

one has that

For the right branch of the hyperbola the range of is:

### Parametric equations

*East-west opening hyperbola:**North-south opening hyperbola:*In all formulae (

*h*,*k*) are the center coordinates of the hyperbola,*a*is the length of the semi-major axis, and*b*is the length of the semi-minor axis.### Elliptic coordinates

A family of confocal hyperbolas is the basis of the system of elliptic coordinatesElliptic coordinates

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in whichthe coordinate lines are confocal ellipses and hyperbolae...

in two dimensions. These hyperbolas are described by the equation

where the foci are located at a distance

*c*from the origin on the*x*-axis, and where θ is the angle of the asymptotes with the*x*-axis. Every hyperbola in this family is orthogonal to every ellipse that shares the same foci. This orthogonality may be shown by a conformal mapConformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

of the Cartesian coordinate system

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

*w*=*z*+ 1/*z*, where*z*=*x*+*iy*are the original Cartesian coordinates, and*w*=*u*+*iv*are those after the transformation.Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings. For example, the mapping

*w*=*z*^{2}transforms the Cartesian coordinate systemCartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

into two families of orthogonal hyperbolas.

## Rectangular hyperbola with horizontal/vertical asymptotes (Cartesian coordinates)

Rectangular hyperbolas with the coordinate axes parallel to their asymptotes have the equation.These are equilateral hyperbolas (eccentricity ) with semi-major axis and semi-minor axis given by .

The simplest example of rectangular hyperbolas occurs when the center (

*h*,*k*) is at the origin:describing quantities

*x*and*y*that are inversely proportional.## Other properties of hyperbolas

- If a line intersects one branch of a hyperbola at M and N and intersects the asymptotes at P and Q, then MN has the same midpoint as PQ.

- The following are concurrentConcurrent linesIn geometry, two or more lines are said to be concurrent if they intersect at a single point.In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:...

: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.

- The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes.

- The product of the distances from a point P to one of the asymptotes along a line parallel to the other asymptote, and to the second asymptote along a line parallel to the first asymptote, is independent of the location of point P on the hyperbola.

- The product of the slopes of lines from a point on the hyperbola to the two vertices is independent of the location of the point.

- A line segment between the two asymptotes and tangent to the hyperbola is bisected by the tangency point.

- The area of a triangle two of whose sides lie on the asymptotes, and whose third side is tangent to the hyperbola, is independent of the location of the tangency point. Specifically, the area is
*ab*, where*a*is the semi-major axis and*b*is the semi-minor axis.

- The distance from either focus to either asymptote is
*b*, the semi-minor axis; the nearest point to a focus on an asymptote lies at a distance from the center equal to*a*, the semi-major axis. Then using the Pythagorean theoremPythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

on the right triangle with these two segments as legs shows that , where*c*is the semi-focal length (the distance from a focus to the hyperbola's center).

### Sundials

Hyperbolas may be seen in many sundialSundial

A sundial is a device that measures time by the position of the Sun. In common designs such as the horizontal sundial, the sun casts a shadow from its style onto a surface marked with lines indicating the hours of the day. The style is the time-telling edge of the gnomon, often a thin rod or a...

s. On any given day, the sun revolves in a circle on the celestial sphere

Celestial sphere

In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...

, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day. The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a

*pelekinon*by the Greeks, since it resembles a double-bladed axe.### Trilateration

A hyperbola is the basis for solving trilaterationTrilateration

In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...

problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from a LORAN

LORAN

LORAN is a terrestrial radio navigation system using low frequency radio transmitters in multiple deployment to determine the location and speed of the receiver....

or GPS transmitters. Conversely, a homing beacon or any transmitter can be located by comparing the arrival times of its signals at two separate receiving stations; such techniques may be used to track objects and people. In particular, the set of possible positions of a point that has a distance difference of 2

*a*from two given points is a hyperbola of vertex separation 2*a*whose foci are the two given points.### Path followed by a particle

The paths followed by any particle in the classical Kepler problemKepler problem

In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force F that varies in strength as the inverse square of the distance r between them. The force may be either attractive or repulsive...

is a conic section

Conic section

. In particular, if the total energy

*E*of the particle is greater than zero (i.e., if the particle is unbound), the path of such a particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering high-energy particles; for example, the Rutherford experimentGeiger-Marsden experiment

The Geiger–Marsden experiment was an experiment to probe the structure of the atom performed by Hans Geiger and Ernest Marsden in 1909, under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester...

demonstrated the existence of an atomic nucleus

Atomic nucleus

The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

by examining the scattering of alpha particle

Alpha particle

Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus, which is classically produced in the process of alpha decay, but may be produced also in other ways and given the same name...

s from gold

Gold

Gold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...

atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb force

Coulomb's law

Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

, which satisfies the inverse square law requirement for a Kepler problem.

### Angle trisection

As shown first by Apollonius of PergaApollonius of Perga

Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

, a hyperbola can be used to trisect any angle

Angle trisection

Angle trisection is a classic problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one-third of a given arbitrary angle, using only two tools: an un-marked straightedge, and a compass....

, an intensely studied problem of geometry. Given an angle, one first draws a circle centered on its middle point

**O**, which intersects the legs of the angle at points**A**and**B**. One next draws the line through**A**and**B**and constructs a hyperbola of eccentricityEccentricity (mathematics)

ε=2 with that line as its transverse axis and

**B**as one focus. The directrix of the hyperbola is the bisector of AB, and for any point**P**on the hyperbola, the angle ABP is twice as large as the angle BAP. Let**P**be a point on the circle. By the inscribed angle theorem, the corresponding center angles are likewise related by a factor of two, AOP = 2×POB. But AOP+POB equals the original angle AOB. Therefore, the angle has been trisected, since 3×POB = AOB.