Definitions

conjugate axes

Hyperbola

[hahy-pur-buh-luh]

In geometry, a hyperbola (Greek ὑπερβολή, "over-thrown") has several equivalent definitions. First, a hyperbola is one of the three types of conic sections, along with ellipses and parabolas; it is a planar open curve resulting from the intersection of a right circular conical surface and a plane that cuts through both halves ("nappes") of a double cone (Figure 1). Second, a hyperbola may also be defined as the locus of points in the plane such that the difference of distances to two fixed points (the foci) is a constant, 2a. Third, a hyperbola may be defined as the locus of points such that the ratio of distances to one focus and to a fixed line (the directrix) equals another constant, the eccentricity ε, which is greater than one. Fourth, a hyperbola may be defined as the reciprocation of a circle in a second circle, where the eccentricity ε equals the distance between the circles' centers divided by the radius of the second circle. Fifth, a hyperbola may be defined as a second-degree (quadratic) equation in the Cartesian coordinates, whose coefficients satisfy two conditions on their determinants.

Similar to an ellipse, a hyperbola can be described in terms of several elements, such as its center, its two foci, its two directrices, its two vertices, its eccentricity, and its principal axes. The hyperbola is an open curve with a center and two branches; both branches gradually approach two lines, known as the hyperbola's asymptotes, that intersect at the hyperbola's center. An ordinary hyperbola never reaches its asymptotes, but a degenerate hyperbola consists only of its asymptotes; the latter occurs when the intersecting plane cuts the cone exactly in half along its axis. The other conic sections, ellipses and parabolas, have no such asymptotes. In either case, the hyperbola has two foci and has a mirror symmetry about two principal axes, the line joining the foci (the transverse axis) and the line perpendicular to it that passes through the center (the conjugate axis). The hyperbola is also symmetrical under a 180° rotation about its center. The two points at which the hyperbola crosses the transverse axis are known as the vertices, which are a distance 2a apart. The transverse and conjugate axes are sometimes called the semi-major and semi-minor axis, respectively.

The hyperbola has several applications, both in mathematics and in other fields. A graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola. The hyperbola led to the definition of the hyperbolic functions that resemble the trigonometric functions defined on the [[circle. Hyperbolas are found in many types of orthogonal coordinates in two and three dimensions, such as the three-dimensional prolate spheroidal coordinates. The shadow of the tip of a sundial usually traces out a hyperbola. The path of a particle being repelled by a center of force is a hyperbola; a classic example is the Rutherford experiment demonstrating the existence of the atomic nucleus, in which alpha particles were scattered from gold atoms.

The hyperbola has several generalizations. Rotation about its conjugate and transverse axes produces a hyperboloid of one and two sheets, respectively.

History

The word "hyperbola" derives from the Greek ὑπερβολή, meaning "over-thrown" or "excessive", from which the English term hyperbole 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 sections, the Conics, although earlier geometers such as Menaechmus and Euclid. For comparison, the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "just right"; 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.

Nomenclature

A hyperbola consists of two disconnected curves called its arms or branches. Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does.

A hyperbola has two focal points (foci). The line connecting these foci is known as the transverse axis, and the midpoint between the foci is known as the hyperbola's center. The line through the center that is perpendicular to the transverse axis is known as the conjugate axis. These axes are known as the two principal axes of the hyperbola. The hyperbola has mirror symmetry about both principal axes, and is also symmetric under a 180° turn about the hyperbola's center. The transverse and conjugate axes are sometimes called the semi-major and semi-minor axis, respectively. The points at which the hyperbola crosses the transverse axis are known as the vertices of the hyperbola, and are located a distance 2a apart; thus, a is the distance from the center to each vertex.

At large distances from the center, the hyperbola approaches two lines, its asymptotes, which intersect at the hyperbola's center. A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them; however, a degenerate hyperbola consists only of its asymptotes. Consistent with the symmetry of the hyperbola, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±b/a; the angle θ between the transverse axis and either asymptote equals arctan(b/a). If b = a, the angle 2θ between the asymptotes equals 90° and the hyperbola is said to rectangular or equilateral. If the transverse axis is aligned with the x-axis of a Cartesian coordinate system, the equation of a hyperbola centered on the origin can be written

frac{x^{2}}{a^{2}} - frac{y^{2}}{b^{2}} = 1

A hyperbola aligned in this way is sometimes called an "East-West opening hyperbola"; by analogy, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola".

The shape of a hyperbola is defined entirely by its eccentricity ε, which is a dimensionless number always greater than one. The distance c from the center to the foci equals aε. 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 equals

epsilon = frac{c}{a} = frac{sqrt{a^{2} + b^{2}}}{a} = sqrt{1 + frac{b^{2}}{a^{2}}} = sec theta

For example, the eccentricity of a rectangular hyperbola (θ = 45°, a = b) equals the square root of two, ε = √2.

Every hyperbola has a conjugate hyperbola, in which the transverse and conjugate axes are exchanged without changing the asymptotes. This corresponds to exchanging a and b in the formulae describing the hyperbola; for example, the angle θ of the conjugate hyperbola equals 90° minus the angle of the original hyperbola. Thus, unless θ 45° (a rectangular hyperbola), the angles in the original and conjugate hyperbolas differ, which implies that they have different eccentricities. Hence, the conjugate hyperbola does not 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 passed 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 2b2/a. The semi-latus rectum l is half of this length, i.e., l = b2/a. The focal parameter p is the distance from a focus to its corresponding directrix, and equals p = l/ε.

Mathematical definitions

A hyperbola can be defined mathematically in several equivalent ways.

Conic section and Dandelin spheres

A hyperbola may be defined as the curve of intersection 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 ellipse and the 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 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 difference of the distances to the two foci is a constant equal to 2a, the distance between its two vertices. This definition accounts for many of the hyperbola's applications, such as trilateration; this is the problem of determining position from the difference in arrival times of synchronized signals, as in GPS.

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 2a 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 2a. 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 F1P F2. 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.

Directrix

The locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola.

Reciprocation of a circle

The reciprocation of a circle 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 polar, 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, then

epsilon = frac{overline{BC}}{r}

Since 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 of the poles of the tangent lines to the circle B, as well as the envelope 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 plane

A_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0

provided that the constants Axx, Axy, Ayy, Bx, By, and C satisfy the determinant condition

D = begin{vmatrix} A_{xx} & A_{xy}A_{xy} & A_{yy} end{vmatrix} < 0,

A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero

Delta := begin{vmatrix} A_{xx} & A_{xy} & B_{x} A_{xy} & A_{yy} & B_{y}B_{x} & B_{y} & C end{vmatrix} = 0

This determinant Δ is sometimes called the discriminant of the conic section.

The center (xc, yc) of the hyperbola may be determined from the formulae

x_{c} = -frac{1}{D} begin{vmatrix} B_{x} & A_{xy} B_{y} & A_{yy} end{vmatrix}

y_{c} = -frac{1}{D} begin{vmatrix} A_{xx} & B_{x} A_{xy} & B_{y} end{vmatrix}

In terms of new coordinates, ξ = x − xc and η = y − yc, the defining equation of the hyperbola can be written

A_{xx} xi^{2} + 2A_{xy} xieta + A_{yy} eta^{2} + frac{Delta}{D} = 0

The principal axes of the hyperbola make an angle Φ with the positive x-axis that equals

tan 2Phi = frac{2A_{xy}}{A_{xx} - A_{yy}}

Rotating the coordinate axes so that the x-axis is aligned with the transverse axis brings the equation into its standard form

frac{{x}^{2}}{a^{2}} - frac{{y}^{2}}{b^{2}} = 1

The major and minor semiaxes a and b are defined by the equations

a^{2} = -frac{Delta}{lambda_{1}D} = -frac{Delta}{lambda_{1}^{2}lambda_{2}}

b^{2} = -frac{Delta}{lambda_{2}D} = -frac{Delta}{lambda_{1}lambda_{2}^{2}}

where λ1 and λ2 are the roots of the quadratic equation

lambda^{2} - left(A_{xx} + A_{yy} right)lambda + D = 0

For comparison, the corresponding equation for a degenerate hyperbola is

frac{{x}^{2}}{a^{2}} - frac{{y}^{2}}{b^{2}} = 0

The tangent line to a given point (x0, y0) on the hyperbola is defined by the equation

E x + F y + G = 0

where E, F and G are defined

E = A_{xx} x_{0} + A_{xy} y_{0} + B_{x}

F = A_{xy} x_{0} + A_{yy} y_{0} + B_{y}

G = B_{x} x_{0} + B_{y} y_{0} + C

The normal line to the hyperbola at the same point is given by the equation

F left(x - x_{0} right) - E left(y - y_{0} right) = 0

The normal line is perpendicular to the tangent line, and both pass through the same point (x0, y0).

Geometrical constructions

Similar to the ellipse, a hyperbola can be constructed using a taut thread. A straightedge of length S is attached to one focus F1 at one of its corners A so that it is free to rotate about that focus. A thread of length L = S - 2a is attached between the other focus F2 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 L2 from F2 to P and LB from P to B. Similarly, the total length S of the straightedge equals the distance L1 from F1 to P and LB. Therefore, the difference in the distances to the foci, L1 − L2 equals the constant 2a

L_{1} - L_{2} = left(S - L_{B} right) - left(L - L_{B} right) = S - L = 2a

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 F1 and F2, 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, QT − PT = 2a, since 2a is the distance between the vertices. Hence, the two circles centered on the foci F1 and F2 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 envelope of the hyperbola.

Geometrical properties

The ancient Greek geometers recognized a reflection property of hyperbolas. If a ray of light emerges from one focus and is reflected 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 ellipses 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 at equal angles; the tangent line to a hyperbola at a point P bisects the angle formed with the two foci, F1PF2.

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 sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola. Assuming that the foci lie at ±c on the x-axis, the Cartesian coordinates of one branch of the hyperbola can be expressed as

x = c costheta cosh mu = a cosh mu

y = c sintheta sinh mu = b sinh mu

where μ is varies over the real numbers, and θ is the angle of the asymptotes with the line of foci. To generate both branches of the hyperbola, the parametric equation can be expressed in terms of the secant and tangent trigonometric functions

x = c costheta sec psi = a sec psi

y = c sintheta tan psi = b tan psi

Relation with other conic sections

There are three major types of conic sections: hyperbolas, ellipses and parabolas. 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 basic equation for an hyperbola

frac{x^{2}}{a^{2}} - frac{y^{2}}{b^{2}} = 1

may be seen as a version of the corresponding ellipse equation

frac{x^{2}}{a^{2}} + frac{y^{2}}{b^{2}} = 1

in which the semi-minor axis length b is imaginary. Similarly, the parametric equations for a hyperbola and an ellipse are expressed in terms of hyperbolic and trigonometric functions, respectively, which are again related by an imaginary number, e.g.,

cosh mu = cos imu

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).

Coordinate systems

A family of confocal hyperbolas is the basis of the system of elliptic coordinates in two dimensions. These hyperbolas are described by the equation

frac{x^{2}}{c^{2} cos^{2}theta} - frac{y^{2}}{c^{2} sin^{2}theta} = 1

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 map of the Cartesian coordinate system 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 = z2 transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.

Applications

Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the a 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, by definition. 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 ax.

A hyperbola is the basis for solving trilateration 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 2a from two given points is a hyperbola of vertex separation 2a whose foci are the two given points.

The paths followed by any particle in the classical Kepler problem is a 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 experiment demonstrated the existence of an atomic nucleus by examining the scattering of alpha particles from gold atoms. If the short-range nuclear interactions are ignoreed, 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.

As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, a 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 eccentricity 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.

Derived curves

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

Extensions

The three-dimensional analog of a hyperbola is a hyperboloid. Hyperboloid come in two varieties, those of one sheet and those of two sheets. A simple way of producing a hyperboloid is to rotate a hyperbola about the axis of its foci or about its symmetry axis perpendicular to the first axis; these rotations produce hyperboloids of two and one sheet, respectively.

Equations for special cases

In Cartesian coordinates

East-west opening hyperbola centered at (h,k):
frac{left(x-h right)^2}{a^2} - frac{left(y-k right)^2}{b^2} = 1
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 a is the semi-major axis (half the distance between the two arms of the hyperbola measured along the major axis), and b is the semi-minor axis.

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the length of 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 is given by

varepsilon = sqrt{1+frac{b^2}{a^2}} = secleft(arctanleft(frac{b}{a}right)right) = coshleft(operatorname{arcsinh}left(frac{b}{a}right)right)

If c equals the distance from the center to either focus, then

varepsilon = frac{c}{a}
where
c = sqrt{a^2 + b^2}.
The distance c is known as the linear eccentricity of the hyperbola. The distance between the foci is 2c or 2.

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

left(hpm c, kright)
and for a north-south opening hyperbola are given by
left(h, kpm cright).

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

x = hpm a ; cosleft(arctanleft(frac{b}{a}right)right)
and for a north-south opening hyperbola are given by
y = kpm a ; cosleft(arctanleft(frac{b}{a}right)right).

Cartesian (rectangular hyperbola with horizontal/vertical asymptotes)

For rectangular hyperbolas with the coordinate axes parallel to their asymptotes:

(x-h)(y-k) = m ,

These are equilateral hyperbolas (eccentricity varepsilon = sqrt 2) with semi-major axis and semi-minor axis given by a=b=sqrt{2m}.

The simplest example of these are the hyperbolas

y=frac{m}{x},.
describing quantities x and y that are inversely proportional.

In polar coordinates

East-west opening hyperbola:
r^2 =asec 2theta ,
North-south opening hyperbola:
r^2 =-asec 2theta ,
Northeast-southwest opening hyperbola:
r^2 =acsc 2theta ,
Northwest-southeast opening hyperbola:
r^2 =-acsc 2theta ,

In all formulas the center is at the pole, and a is the semi-major axis and semi-minor axis.

Parametric equations

East-west opening hyperbola:
begin{matrix}
x = asec t + h 
y = btan t + k 
end{matrix} qquad mathrm{or} qquadbegin{matrix}
x = pm acosh t + h 
y = bsinh t + k 
end{matrix}

North-south opening hyperbola:

begin{matrix}
x = atan t + h 
y = bsec t + k 
end{matrix} qquad mathrm{or} qquadbegin{matrix}
x = asinh t + h 
y = pm bcosh t + k 
end{matrix}

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.

See also

External links

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