Hyperbola

This article is about a geometric curve. For the term usedin rhetoric, see Hyperbole.

In mathematics, a hyperbola (plural hyperbolas or

A hyperbola is an open curve with two branches, the intersectionof a plane with both halves of a double cone. The plane does nothave to be parallel to the axis of the cone; the hyperbola will besymmetrical in any case.

hyperbolae) is a type of smooth curve, lying in a plane,defined by its geometric properties or by equations forwhich it is the solution set. A hyperbola has two pieces,called connected components or branches, that are mir-ror images of each other and resemble two infinite bows.The hyperbola is one of the four kinds of conic section,formed by the intersection of a plane and a double cone.(The other conic sections are the parabola, the ellipse, andthe circle; the circle is a special case of the ellipse). If theplane intersects both halves of the double cone but doesnot pass through the apex of the cones, then the conic isa hyperbola.Hyperbolas arise in many ways: as the curve represent-

Hyperbolas in the physical world: three cones of light of differentwidths and intensities are generated by a (roughly) downwards-pointing halogen lamp and its housing. Each cone of light inter-sects a nearby vertical wall in a hyperbola.

ing the function f(x) = 1/x in the Cartesian plane, asthe appearance of a circle viewed from within it, as thepath followed by the shadow of the tip of a sundial, asthe shape of an open orbit (as distinct from a closed el-liptical orbit), such as the orbit of a spacecraft during agravity assisted swing-by of a planet or more generallyany spacecraft exceeding the escape velocity of the near-est planet, as the path of a single-apparition comet (onetravelling too fast ever to return to the solar system), asthe scattering trajectory of a subatomic particle (acted onby repulsive instead of attractive forces but the principleis the same), and so on.Each branch of the hyperbola has two arms which be-come straighter (lower curvature) further out from thecenter of the hyperbola. Diagonally opposite arms, onefrom each branch, tend in the limit to a common line,called the asymptote of those two arms. So there are twoasymptotes, whose intersection is at the center of sym-metry of the hyperbola, which can be thought of as themirror point about which each branch reflects to form the

1

2 2 NOMENCLATURE AND FEATURES

Hyperbolas produced by interference of waves

other branch. In the case of the curve f(x) = 1/x theasymptotes are the two coordinate axes.Hyperbolas share many of the ellipses’ analytical proper-ties such as eccentricity, focus, and directrix. Typicallythe correspondence can be made with nothing more thana change of sign in some term. Many other mathematicalobjects have their origin in the hyperbola, such ashyperbolic paraboloids (saddle surfaces), hyperboloids(“wastebaskets”), hyperbolic geometry (Lobachevsky'scelebrated non-Euclidean geometry), hyperbolic func-tions (sinh, cosh, tanh, etc.), and gyrovector spaces (a ge-ometry used in both relativity and quantum mechanicswhich is not Euclidean).

1 History

The word “hyperbola” derives from the Greek ὑπερβολή,meaning “over-thrown” or “excessive”, from which theEnglish term hyperbole also derives. Hyperbolae werediscovered by Menaechmus in his investigations of theproblem of doubling the cube, but were then called sec-tions of obtuse cones.[1] The term hyperbola is believed tohave been coined by Apollonius of Perga (c. 262–c. 190BC) in his definitive work on the conic sections, the Con-ics.[2] For comparison, the other two general conic sec-tions, the ellipse and the parabola, derive from the corre-sponding 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 (el-lipse) and exactly one (parabola).

2 Nomenclature and features

F2 F1

D1D2

P

e·PD1

e>1

a−a

a/e

C

ae

The hyperbola consists of the red curves. The asymptotes of thehyperbola are shown as blue dashed lines and intersect at thecenter of the hyperbola, C. The two focal points are labeled F1

and F2, and the thin black line joining them is the transverse axis.The perpendicular thin black line through the center is the con-jugate axis. The two thick black lines parallel to the conjugateaxis (thus, perpendicular to the transverse axis) are the two di-rectrices, D1 and D2. The eccentricity e equals the ratio of thedistances from a point P on the hyperbola to one focus and itscorresponding directrix line (shown in green). The two verticesare located on the transverse axis at ±a relative to the center. Sothe parameters are: a— distance from center C to either vertexb — length of a segment perpendicular to the transverse axisdrawn from each vertex to the asymptotesc — distance from center C to either Focus point, F1 and F2,andθ — angle formed by each asymptote with the transverse axis.

Similar to a parabola, a hyperbola is an open curve, mean-ing that it continues indefinitely to infinity, rather thanclosing on itself as an ellipse does. A hyperbola consistsof two disconnected curves called its arms or branches.The points on the two branches that are closest to eachother are called the vertices; they are the points where thecurve has its smallest radius of curvature. The line seg-ment connecting the vertices is called the transverse axisor major axis, corresponding to the major diameter of anellipse. The midpoint of the transverse axis is known asthe hyperbola’s center. The distance a from the centerto each vertex is called the semi-major axis. Outside ofthe transverse axis but on the same line are the two focalpoints (foci) of the hyperbola. The line through these fivepoints is one of the two principal axes of the hyperbola,the other being the perpendicular bisector of the trans-verse axis. The hyperbola has mirror symmetry about itsprincipal axes, and is also symmetric under a 180° turnabout its center.At large distances from the center, the hyperbola ap-proaches two lines, its asymptotes, which intersect at thehyperbola’s center. A hyperbola approaches its asymp-

3

totes arbitrarily closely as the distance from its cen-ter increases, but it never intersects them; however, adegenerate hyperbola consists only of its asymptotes.Consistent with the symmetry of the hyperbola, if thetransverse axis is aligned with the x-axis of a Cartesiancoordinate system, the slopes of the asymptotes are equalin magnitude but opposite in sign, ±b⁄ₐ, where b=a×tan(θ)and where θ is the angle between the transverse axis andeither asymptote. The distance b (not shown) is the lengthof the perpendicular segment from either vertex to theasymptotes.A conjugate axis of length 2b, corresponding to the mi-nor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the mi-nor axis at the height of the asymptotes over/under thehyperbola’s vertices. Because of the minus sign in someof the formulas below, it is also called the imaginary axisof the hyperbola.If b = a, 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 pointson the asymptotes directly above and below the verticesis a square, since the lengths of its sides 2a = 2b.If the transverse axis of any hyperbola is aligned with thex-axis of a Cartesian coordinate system and is centered onthe origin, the equation of the hyperbola can be writtenas

x2

a2− y2

b2= 1.

A hyperbola aligned in this way is called an “East-Westopening hyperbola”. Likewise, a hyperbola with its trans-verse axis alignedwith the y-axis is called a “North–Southopening hyperbola” and has equation

y2

a2− x2

b2= 1.

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 tothe origin-centered North–South opening hyperbola withidentical eccentricity ε — that is, it can be rotated so thatit opens in the desired direction and can be translated(rigidly moved in the plane) so that it is centered at theorigin. For convenience, hyperbolas are usually analyzedin terms of their centered East-West opening form.If c is the distance from the center to either focus, thena2 + b2 = c2 .The shape of a hyperbola is defined entirely by itseccentricity ε, which is a dimensionless number alwaysgreater than one. The distance c from the center to thefoci equals aε. The eccentricity can also be defined asthe ratio of the distances to either focus and to a corre-sponding line known as the directrix; hence, the distance

−4 −3 −2 −1 0 4 321

1

2

3

4

−1

−2

−3

−4

y

x

x2−y2=−1 x2−y2=1

asymptotes

Here a = b = 1 giving the unit hyperbola in blue and its conjugatehyperbola in green, sharing the same red asymptotes.

from the center to the directrices equals a/ε. In terms ofthe parameters a, b, c and the angle θ, the eccentricityequals

ε =c

a=

√a2 + b2

a=

√1 +

b2

a2= sec θ.

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 whichthe transverse and conjugate axes are exchanged withoutchanging the asymptotes. The equation of the conjugatehyperbola of x2

a2 − y2

b2 = 1 is x2

a2 − y2

b2 = −1 . If the graphof the conjugate hyperbola is rotated 90° to restore theeast-west opening orientation (so that x becomes y andvice versa), the equation of the resulting rotated conju-gate hyperbola is the same as the equation of the originalhyperbola except with a and b exchanged. For example,the angle θ of the conjugate hyperbola equals 90° minusthe angle of the original hyperbola. Thus, the angles inthe original and conjugate hyperbolas are complementaryangles, which implies that they have different eccentrici-ties unless θ = 45° (a rectangular hyperbola). Hence, theconjugate hyperbola does not in general correspond to a90° rotation of the original hyperbola; the two hyperbolasare 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 ofthe hyperbola’s foci. The line segment connecting thetwo intersection points of this line with the hyperbola isknown as the latus rectum and has a length 2b2

a . Thesemi-latus rectum l is half of this length, i.e., l = b2

a .The focal parameter p is the distance from a focus to itscorresponding directrix, and equals p = b2

c .

4 3 MATHEMATICAL DEFINITIONS

3 Mathematical definitions

A hyperbola can be defined mathematically in severalequivalent ways.

3.1 Conic section

Three major types of conic sections.

A hyperbola may be defined as the curve of intersectionbetween a right circular conical surface and a plane thatcuts through both halves of the cone. The other majortypes of conic sections are the ellipse and the parabola;in these cases, the plane cuts through only one half of thedouble cone. If the plane passes through the central apexof the double cone a degenerate hyperbola results — twostraight lines that cross at the apex point.

3.2 Difference of distances to foci

A hyperbola may be defined equivalently as the locus ofpoints where the absolute value of the difference of thedistances to the two foci is a constant equal to 2a, thedistance between its two vertices. This definition ac-counts for many of the hyperbola’s applications, such asmultilateration; this is the problem of determining posi-tion from the difference in arrival times of synchronizedsignals, as in GPS.This definition may be expressed also in terms of tangentcircles. The center of any circles externally tangent totwo given circles lies on a hyperbola, whose foci are thecenters of the given circles and where the vertex distance2a equals the difference in radii of the two circles. As aspecial case, one given circle may be a point located atone focus; since a point may be considered as a circle ofzero radius, the other given circle—which is centered onthe other focus—must have radius 2a. This provides asimple technique for constructing a hyperbola, as shownbelow. It follows from this definition that a tangent line tothe hyperbola at a point P bisects the angle formed withthe two foci, i.e., the angle F1P F2. Consequently, thefeet of perpendiculars drawn from each focus to such atangent line lies on a circle of radius a that is centered onthe hyperbola’s own center.

A proof that this characterization of the hyperbola isequivalent to the conic-section characterization can bedone without coordinate geometry by means of Dandelinspheres.

3.3 Directrix and focus

A hyperbola can be defined as the locus of points forwhich the ratio of the distances to one focus and to a line(called the directrix) is a constant ϵ that is larger than 1.This constant is the eccentricity of the hyperbola. Theeccentricity equals the secant of half the angle betweenthe asymptotes of the hyperbola, so the eccentricity ofthe hyperbola xy = 1 equals the square root of 2.By symmetry a hyperbola has two directrices, which areparallel to the conjugate axis and are between it and thetangent to the hyperbola at a vertex. One directrix and itsfocus is enough to produce both arms of the hyperbola.

3.4 Reciprocation of a circle

The reciprocation of a circle B in a circle C always yieldsa conic section such as a hyperbola. The process of “re-ciprocation in a circle C" consists of replacing every lineand point in a geometrical figure with their correspond-ing pole and polar, respectively. The pole of a line is theinversion of its closest point to the circle C, whereas thepolar of a point is the converse, namely, a line whose clos-est point to C is the inversion of the point.The eccentricity of the conic section obtained by recipro-cation is the ratio of the distances between the two circles’centers to the radius r of reciprocation circle C. If B andC represent the points at the centers of the correspondingcircles, then

ϵ =BC

r

Since the eccentricity of a hyperbola is always greaterthan one, the center B must lie outside of the recipro-cating circle C.This definition implies that the hyperbola is both thelocus of the poles of the tangent lines to the circle B, aswell as the envelope of the polar lines of the points on B.Conversely, the circle B is the envelope of polars of pointson the hyperbola, and the locus of poles of tangent linesto the hyperbola. Two tangent lines to B have no (finite)poles because they pass through the centerC of the recip-rocation circle C; the polars of the corresponding tangentpoints on B are the asymptotes of the hyperbola. The twobranches of the hyperbola correspond to the two parts ofthe circle B that are separated by these tangent points.

3.5 Quadratic equation 5

3.5 Quadratic equation

A hyperbola can also be defined as a second-degree equa-tion in the Cartesian coordinates (x, y) of the plane

Axxx2 + 2Axyxy +Ayyy

2 + 2Bxx+ 2Byy + C = 0

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

D =

∣∣∣∣Axx Axy

Axy Ayy

∣∣∣∣ < 0

A special case of a hyperbola—the degenerate hyperbolaconsisting of two intersecting lines—occurs when anotherdeterminant is zero

∆ :=

∣∣∣∣∣∣Axx Axy Bx

Axy Ayy By

Bx By C

∣∣∣∣∣∣ = 0

This determinant Δ is sometimes called the discriminantof the conic section.[3]

Given the above general parametrization of the hyperbolain Cartesian coordinates, the eccentricity can be foundusing the formula in Conic section#Eccentricity in termsof parameters of the quadratic form.The center (xc, yc) of the hyperbola may be determinedfrom the formulae

xc = − 1

D

∣∣∣∣Bx Axy

By Ayy

∣∣∣∣yc = − 1

D

∣∣∣∣Axx Bx

Axy By

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

Axxξ2 + 2Axyξη +Ayyη

2 +∆

D= 0

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

tan 2Φ =2Axy

Axx −Ayy

Rotating the coordinate axes so that the x-axis is alignedwith the transverse axis brings the equation into itscanonical form

x2

a2− y2

b2= 1

The major and minor semiaxes a and b are defined by theequations

a2 = − ∆

λ1D= − ∆

λ21λ2

b2 = − ∆

λ2D= − ∆

λ1λ22

where λ1 and λ2 are the roots of the quadratic equation

λ2 − (Axx +Ayy)λ+D = 0

For comparison, the corresponding equation for a degen-erate hyperbola is

x2

a2− y2

b2= 0

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

Ex+ Fy +G = 0

where E, F and G are defined

E = Axxx0 +Axyy0 +Bx

F = Axyx0 +Ayyy0 +By

G = Bxx0 +Byy0 + C

The normal line to the hyperbola at the same point is givenby the equation

F (x− x0)− E (y − y0) = 0

The normal line is perpendicular to the tangent line, andboth pass through the same point (x0, y0).From the equation

x2

a2− y2

b2= 1 0 < b ≤ a

the basic property that with r1 and r2 being the distancesfrom a point (x, y) to the left focus (−ae, 0) and the rightfocus (ae, 0) one has for a point on the right branch that

r1 − r2 = 2a

and for a point on the left branch that

r2 − r1 = 2a

6 5 GEOMETRICAL CONSTRUCTIONS

can be proved as follows:If x,y is a point on the hyperbola the distance to the leftfocal point is

r21 = (x+ae)2+y2 = x2+2xae+a2e2+(x2−a2)(e2−1) = (ex+a)2

To the right focal point the distance is

r22 = (x−ae)2+y2 = x2−2xae+a2e2+(x2−a2)(e2−1) = (ex−a)2

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

r1 = ex+ a

r2 = ex− a

Subtracting these equations one gets

r1 − r2 = 2a

If x,y is a point on the left branch of the hyperbola thenex < −a and

r1 = −ex− a

r2 = −ex+ a

Subtracting these equations one gets

r2 − r1 = 2a

4 True anomaly

In the section above it is shown that using the coordinatesystem in which the equation of the hyperbola takes itscanonical form

x2

a2− y2

b2= 1

the distance r from a point (x , y) on the left branch ofthe hyperbola to the left focal point (−ea , 0) is

r = −ex− a

Introducing polar coordinates (r , θ) with origin at theleft focal point the coordinates relative the canonical co-ordinate system are

r

Focus

The angle shown is the true anomaly of the indicated point on thehyperbola.

x = −ae+ r cos θ

y = r sin θand the equation above takes the form

r = −e(−ae+ r cos θ)− a

from which follows that

r =a(e2 − 1)

1 + e cos θThis is the representation of the near branch of a hyper-bola in polar coordinates with respect to a focal point.The polar angle θ of a point on a hyperbola relative thenear focal point as described above is called the trueanomaly of the point.

5 Geometrical constructions

Similar to the ellipse, a hyperbola can be constructed us-ing a taut thread. A straightedge of length S is attachedto one focus F1 at one of its corners A so that it is free to

7

Hyperbola construction using the parallelogram method

rotate about that focus. A thread of length L = S - 2a isattached between the other focus F2 and the other cornerB of the straightedge. A sharp pencil is held up againstthe straightedge, sandwiching the thread tautly against thestraightedge. Let the position of the pencil be denoted asP. The total length L of the thread equals the sum of thedistances L2 from F2 to P and LB from P to B. Similarly,the total length S of the straightedge equals the distanceL1 from F1 to P and LB. Therefore, the difference in thedistances to the foci, L1 − L2 equals the constant 2a

L1 − L2 = (S − LB)− (L− LB) = S − L = 2a

A second construction uses intersecting circles, but islikewise based on the constant difference of distances tothe foci. Consider a hyperbola with two foci F1 and F2,and two vertices P and Q; these four points all lie on thetransverse axis. Choose a new point T also on the trans-verse axis and to the right of the rightmost vertex P; thedifference in distances to the two vertices, QT − PT = 2a,since 2a is the distance between the vertices. Hence, thetwo circles centered on the foci F1 and F2 of radius QTand PT, respectively, will intersect at two points of thehyperbola.A third construction relies on the definition of the hyper-bola as the reciprocation of a circle. Consider the circlecentered on the center of the hyperbola and of radius a;this circle is tangent to the hyperbola at its vertices. A lineg drawn from one focus may intersect this circle in twopointsM and N; perpendiculars to g drawn through thesetwo points are tangent to the hyperbola. Drawing a set ofsuch tangent lines reveals the envelope of the hyperbola.A fourth construction is using the parallelogram method.It is similar to such method for parabola and ellipse con-struction: certain equally spaced points lying on parallellines are connected with each other by two straight lines

and their intersection point lies on the hyperbola.

6 Reflections and tangent lines

The ancient Greek geometers recognized a reflectionproperty of hyperbolas. If a ray of light emerges fromone focus and is reflected from either branch of the hyper-bola, the light-ray appears to have come from the other fo-cus. Equivalently, by reversing the direction of the light,rays directed at one of the foci are reflected towards theother focus. This property is analogous to the propertyof ellipses that a ray emerging from one focus is reflectedfrom the ellipse directly towards the other focus (ratherthan away as in the hyperbola). Expressed mathemati-cally, lines drawn from each focus to the same point onthe hyperbola intersect it at equal angles; the tangent lineto a hyperbola at a point P bisects the angle formed withthe two foci, F1PF2.Tangent lines to a hyperbola have another remarkable ge-ometrical property. If a tangent line at a pointT intersectsthe asymptotes at two points K and L, then T bisects theline segment KL, and the product of distances to the hy-perbola’s center, OK×OL is a constant.

7 Hyperbolic functions and equa-tions

Just as the sine and cosine functions give a parametricequation for the ellipse, so the hyperbolic sine andhyperbolic cosine give a parametric equation for the hy-perbola.Ascosh2 µ− sinh2 µ = 1

one has for any hyperbolic angle µ that the point

x = a cosh µ

y = b sinh µ

satisfies the equation

x2

a2− y2

b2= 1

which is the equation of a hyperbola relative its canonicalcoordinate system.When μ varies over the interval −∞ < µ < ∞ one getswith this formula all points (x , y) on the right branch ofthe hyperbola.The left branch for which x < 0 is in the same way ob-tained as

8 9 CONIC SECTION ANALYSIS OF THE HYPERBOLIC APPEARANCE OF CIRCLES

k = -5

k = -4

k = -3

k = -2

k = -1

k = 0

k = 1

k = 2

k = 3

k = 4

k = 5

The points (−a cosh µk , b sinh µk) with µk = 0.3 k fork = −5,−4, · · · , 5

x = −a cosh µ

y = b sinh µ

In the figure the points (xk , yk) given by

xk = −a coshµk

yk = b sinhµk

for

µk = 0.3 k k = −5,−4, · · · , 5

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

8 Relation to other conic sections

There are threemajor types of conic sections: hyperbolas,ellipses and parabolas. Since the parabola may be seen

as a limiting case poised exactly between an ellipse anda hyperbola, there are effectively only two major types,ellipses and hyperbolas. These two types are related inthat formulae for one type can often be applied to theother.The canonical equation for a hyperbola is

x2

a2− y2

b2= 1.

Any hyperbola can be rotated so that it is east-west open-ing and positioned with its center at the origin, so that theequation describing it is this canonical equation.The canonical equation for the hyperbola may be seen asa version of the corresponding ellipse equation

x2

a2+

y2

b2= 1

in which the semi-minor axis length b is imaginary. Thatis, if in the ellipse equation b is replaced by ib where b isreal, one obtains the hyperbola equation.Similarly, the parametric equations for a hyperbola andan ellipse are expressed in terms of hyperbolic andtrigonometric functions, respectively, which are again re-lated by an imaginary circular angle, for example,

coshµ = cos iµ

Hence, many formulae for the ellipse can be extendedto hyperbolas by adding the imaginary unit i in front ofthe semi-minor axis b and the angle. For example, thearc length of a segment of an ellipse can be determinedusing an incomplete elliptic integral of the second kind.The corresponding arclength of a hyperbola is given bythe same function with imaginary parameters b and μ,namely, ib E(iμ, c).

9 Conic section analysis of the hy-perbolic appearance of circles

Besides providing a uniform description of circles, el-lipses, parabolas, and hyperbolas, conic sections can alsobe understood as a natural model of the geometry of per-spective in the case where the scene being viewed consistsof a circle, ormore generally an ellipse. The viewer is typ-ically a camera or the human eye. In the simplest case theviewer’s lens is just a pinhole; the role of more complexlenses is merely to gather far more light while retaining asfar as possible the simple pinhole geometry in which allrays of light from the scene pass through a single point.Once through the lens, the rays then spread out again, inair in the case of a camera, in the vitreous humor in the

9

Figure 10: The hyperbola as a circle on the ground seen in per-spective while gazing down slightly, showing the circle’s (non-parallel) tangents as asymptotes. The portion above the horizonis normally invisible. When gazing straight ahead the tangentswill be parallel and therefore intersect at the horizon instead ofabove, described further in the text.

case of the eye, eventually distributing themselves overthe film, imaging device, or retina, all of which come un-der the heading of image plane. The lens plane is a planeparallel to the image plane at the lens; all rays pass througha single point on the lens plane, namely the lens itself.When the circle directly faces the viewer, the viewer’slens is on-axis, meaning on the line normal to the cir-cle through its center (think of the axle of a wheel). Therays of light from the circle through the lens to the imageplane then form a cone with circular cross section whoseapex is the lens. The image plane concretely realizes theabstract cutting plane in the conic section model.When in addition the viewer directly faces the circle, thecircle is rendered faithfully on the image plane withoutperspective distortion, namely as a scaled-down circle.When the viewer turns attention or gaze away from thecenter of the circle the image plane then cuts the cone inan ellipse, parabola, or hyperbola depending on how farthe viewer turns, corresponding exactly to what happenswhen the surface cutting the cone to form a conic sectionis rotated.A parabola arises when the lens plane is tangent to(touches) the circle. A viewer with perfect 180-degreewide-angle vision will see the whole parabola; in practicethis is impossible and only a finite portion of the parabolais captured on the film or retina.When the viewer turns further so that the lens plane cutsthe circle in two points, the shape on the image plane be-comes that of a hyperbola. The viewer still sees only afinite curve, namely a portion of one branch of the hy-perbola, and is unable to see the second branch at all,

which corresponds to the portion of the circle behind theviewer, more precisely, on the same side of the lens planeas the viewer. In practice the finite extent of the imageplane makes it impossible to see any portion of the circlenear where it is cut by the lens plane. Further back how-ever one could imagine rays from the portion of the cir-cle well behind the viewer passing through the lens, werethe viewer transparent. In this case the rays would passthrough the image plane before the lens, yet another im-practicality ensuring that no portion of the second branchcould possibly be visible.The tangents to the circle where it is cut by the lens planeconstitute the asymptotes of the hyperbola. Were thesetangents to be drawn in ink in the plane of the circle,the eye would perceive them as asymptotes to the visi-ble branch. Whether they converge in front of or behindthe viewer depends on whether the lens plane is in frontof or behind the center of the circle respectively.If the circle is drawn on the ground and the viewer gradu-ally transfers gaze from straight down at the circle up to-wards the horizon, the lens plane eventually cuts the circleproducing first a parabola then a hyperbola on the imageplane as shown in Figure 10. As the gaze continues to risethe asymptotes of the hyperbola, if realized concretely,appear coming in from left and right, swinging towardseach other and converging at the horizon when the gazeis horizontal. Further elevation of the gaze into the skythen brings the point of convergence of the asymptotestowards the viewer.By the same principle with which the back of the circleappears on the image plane were all the physical obstaclesto its projection to be overcome, the portion of the twotangents behind the viewer appear on the image plane asan extension of the visible portion of the tangents in frontof the viewer. Like the second branch this extension ma-terializes in the sky rather than on the ground, with thehorizonmarking the boundary between the physically vis-ible (scene in front) and invisible (scene behind), and thevisible and invisible parts of the tangents combining in asingle X shape. As the gaze is raised and lowered aboutthe horizon, the X shape moves oppositely, lowering asthe gaze is raised and vice versa but always with the visi-ble portion being on the ground and stopping at the hori-zon, with the center of the X being on the horizon whenthe gaze is horizontal.All of the above was for the case when the circle facesthe viewer, with only the viewer’s gaze varying. When thecircle starts to face away from the viewer the viewer’s lensis no longer on-axis. In this case the cross section of thecone is no longer a circle but an ellipse (never a parabolaor hyperbola). However the principle of conic sectionsdoes not depend on the cross section of the cone beingcircular, and applies without modification to the case ofeccentric cones.It is not difficult to see that even in the off-axis case acircle can appear circular, namely when the image plane

10 11 COORDINATE SYSTEMS

(and hence lens plane) is parallel to the plane of the cir-cle. That is, to see a circle as a circle when viewing itobliquely, look not at the circle itself but at the plane inwhich it lies. From this it can be seen that when viewing aplane filled with many circles, all of them will appear cir-cular simultaneously when the plane is looked at directly.A common misperception about the hyperbola is that it isa mathematical curve rarely if ever encountered in dailylife. The reality is that one sees a hyperbola whenevercatching sight of portion of a circle cut by one’s lens plane(and a parabola when the lens plane is tangent to, i.e. justtouches, the circle). The inability to see very much of thearms of the visible branch, combined with the completeabsence of the second branch, makes it virtually impossi-ble for the human visual system to recognize the connec-tion with hyperbolas such as y = 1/x where both branchesare on display simultaneously.

10 Derived curves

Several other curves can be derived from the hyperbola byinversion, the so-called inverse curves of the hyperbola.If the center of inversion is chosen as the hyperbola’s owncenter, the inverse curve is the lemniscate of Bernoulli;the lemniscate is also the envelope of circles centered ona rectangular hyperbola and passing through the origin. Ifthe center of inversion is chosen at a focus or a vertex ofthe hyperbola, the resulting inverse curves are a limaçonor a strophoid, respectively.

11 Coordinate systems

11.1 Cartesian coordinates

An east-west opening hyperbola centered at (h,k) has theequation

(x− h)2

a2− (y − k)

2

b2= 1.

The major axis runs through the center of the hyperbolaand intersects both arms of the hyperbola at the vertices(bend points) of the arms. The foci lie on the extensionof the major axis of the hyperbola.The minor axis runs through the center of the hyperbolaand is perpendicular to the major axis.In both formulas a is the semi-major axis (half the dis-tance between the two arms of the hyperbola measuredalong themajor axis),[4] and b is the semi-minor axis (halfthe distance between the asymptotes along a line tangentto the hyperbola at a vertex).If one forms a rectangle with vertices on the asymptotesand two sides that are tangent to the hyperbola, the sides

n = −2 n = −1 n = −1/2 n = 1/2 n = 1 n = 2

r = 1

r = 2

r = 3

θ = 0

θ = π / 4

θ = π / 2

θ = 3π / 4

θ = π

θ = 5π / 4

θ = 3π / 2

θ = 7π / 4

π_4

π_2

3π__4

π 5π__4

3π__2

7π__4

2πθ

r

3

2

1

0

−1

−2

−3

Sinusoidal spirals: equilateral hyperbola (n = −2), line (n =−1), parabola (n =−1/2), cardioid (n = 1/2), circle (n = 1) andlemniscate of Bernoulli (n = 2), where rn = 1n cos(nθ) in polarcoordinates and their equivalents in rectangular coordinates.

tangent to the hyperbola are 2b in length while the sidesthat run parallel to the line between the foci (the majoraxis) are 2a in length. Note that b may be larger than adespite the names minor and major.If one calculates the distance from any point on the hy-perbola to each focus, the absolute value of the differenceof those two distances is always 2a.The eccentricity is given by

ε =

√1 +

b2

a2= sec

(arctan

(b

a

))= cosh

(arcsinh

(b

a

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

ε =c

a

11.3 Parametric equations 11

where

c =√

a2 + b2

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

(h± c, k)

and for a north-south opening hyperbola are given by

(h, k ± c)

The directrices for an east-west opening hyperbola aregiven by

x = h± a cos(arctan

(b

a

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

y = k ± a cos(arctan

(b

a

))

11.2 Polar coordinates

The polar coordinates used most commonly for the hy-perbola are defined relative to the Cartesian coordinatesystem that has its origin in a focus and its x-axis pointingtowards 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

r =a(e2 − 1)

1 + e cos θ

and the range of the true anomaly θ is:

− arccos(−1

e

)< θ < arccos

(−1

e

)With polar coordinate relative to the “canonical coordi-nate system”

x = R cos t

y = R sin tone has that

R2 =b2

e2 cos2 t− 1

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

− arccos(1

e

)< t < arccos

(1

e

)

11.3 Parametric equations

East-west opening hyperbola:

x = a sec t+ hy = b tan t+ k

or x = ±a cosh t+ hy = b sinh t+ k

North-south opening hyperbola:

x = b tan t+ hy = a sec t+ k

or x = b sinh t+ hy = ±a cosh t+ k

In all formulae (h,k) are the center coordinates of the hy-perbola, a is the length of the semi-major axis, and b isthe length of the semi-minor axis.

11.4 Elliptic coordinates

A family of confocal hyperbolas is the basis of the systemof elliptic coordinates in two dimensions. These hyper-bolas are described by the equation

( x

c cos θ)2

−( y

c sin θ)2

= 1

where the foci are located at a distance c from the ori-gin on the x-axis, and where θ is the angle of the asymp-totes with the x-axis. Every hyperbola in this family isorthogonal to every ellipse that shares the same foci. Thisorthogonality may be shown by a conformal map of theCartesian coordinate system w = z + 1/z, where z= x + iyare the original Cartesian coordinates, and w=u + iv arethose after the transformation.Other orthogonal two-dimensional coordinate systems in-volving hyperbolas may be obtained by other conformalmappings. For example, the mapping w = z2 transformsthe Cartesian coordinate system into two families of or-thogonal hyperbolas.

12 Rectangular hyperbola

A rectangular hyperbola, equilateral hyperbola, orright hyperbola is a hyperbola for which the asymptotesare perpendicular.[5]

12 14 APPLICATIONS

A graph of the rectangular hyperbola y = 1x, the reciprocal

function

Rectangular hyperbolas with the coordinate axes parallelto their asymptotes have the equation

(x− h)(y − k) = m

Rectangular hyperbolas have eccentricity ε =√2 with

semi-major axis and semi-minor axis given by a = b =√2m .

The simplest example of rectangular hyperbolas occurswhen the center (h, k) is at the origin:

y =m

x

describing quantities x and y that are inversely propor-tional. By rotating the coordinate axes counterclockwiseby 45 degrees, with the new coordinate axes labelled(x′, y′) the equation of the hyperbola is given by canoni-cal form

(x′)2

(√2m)2

− (y′)2

(√2m)2

= 1

If the scale factor m=1/2, then this canonical rectangularhyperbola is the unit hyperbola.A circumconic passing through the orthocenter of atriangle is a rectangular hyperbola.[6]

13 Other properties of hyperbolas• If a line intersects one branch of a hyperbola at Mand N and intersects the asymptotes at P and Q, thenMN has the same midpoint as PQ.[7][8]:p.49,ex.7

• The following are concurrent: (1) a circle passingthrough the hyperbola’s foci and centered at the hy-perbola’s center; (2) either of the lines that are tan-gent to the hyperbola at the vertices; and (3) eitherof the asymptotes of the hyperbola.[7][9]

• The following are also concurrent: (1) the circle thatis centered at the hyperbola’s center and that passesthrough the hyperbola’s vertices; (2) either directrix;and (3) either of the asymptotes.[9]

• The product of the distances from a point P on thehyperbola to one of the asymptotes along a lineparallel to the other asymptote, and to the secondasymptote along a line parallel to the first asymp-tote, is independent of the location of point P on thehyperbola.[9] If the hyperbola is written in canonicalform x2

a2 − y2

b2 = 1 then this product is a2+b2

4 .

• The product of the perpendicular distances from apoint P on the hyperbola x2

a2 − y2

b2 = 1 or on its con-jugate hyperbola x2

a2 − y2

b2 = −1 to the asymptotes isa constant independent of the location of P: specif-ically, a2b2

a2+b2 , which also equals (b/e)2 where e isthe eccentricity of the hyperbola x2

a2 − y2

b2 = 1 .[10]

• The product of the slopes of lines from a point onthe hyperbola to the two vertices is independent ofthe location of the point.[11]

• A line segment between the two asymptotes and tan-gent to the hyperbola is bisected by the tangencypoint.[8]:p.49,ex.6[11][12]

• The area of a triangle two of whose sides lie on theasymptotes, and whose third side is tangent to thehyperbola, is independent of the location of the tan-gency point.[8]:p.49,ex.6 Specifically, the area is ab,where a is the semi-major axis and b is the semi-minor axis.[12]

• The distance from either focus to either asymptote isb, the semi-minor axis; the nearest point to a focuson an asymptote lies at a distance from the centerequal to a, the semi-major axis.[7] Then using thePythagorean theorem on the right triangle with thesetwo segments as legs shows that a2+b2 = c2 , wherec is the semi-focal length (the distance from a focusto the hyperbola’s center).

14 Applications

14.1 Sundials

Hyperbolas may be seen in many sundials. On any givenday, the sun revolves in a circle on the celestial sphere, and

14.4 Korteweg-de Vries equation 13

Hyperbolas as declination lines on a sundial

its rays striking the point on a sundial traces out a cone oflight. The intersection of this cone with the horizontalplane of the ground forms a conic section. At most pop-ulated latitudes and at most times of the year, this conicsection is a hyperbola. In practical terms, the shadow ofthe tip of a pole traces out a hyperbola on the ground overthe course of a day (this path is called the declination line).The shape of this hyperbola varies with the geographicallatitude and with the time of the year, since those factorsaffect the cone of the sun’s rays relative to the horizon.The collection of such hyperbolas for a whole year at agiven location was called a pelekinon by the Greeks, sinceit resembles a double-bladed axe.

14.2 Multilateration

A hyperbola is the basis for solving Multilateration prob-lems, the task of locating a point from the differences inits distances to given points — or, equivalently, the dif-ference in arrival times of synchronized signals betweenthe point and the given points. Such problems are impor-tant in navigation, particularly on water; a ship can locateits position from the difference in arrival times of signalsfrom a LORAN or GPS transmitters. Conversely, a hom-ing beacon or any transmitter can be located by compar-ing the arrival times of its signals at two separate receiv-ing stations; such techniques may be used to track objectsand people. In particular, the set of possible positions ofa point that has a distance difference of 2a from two givenpoints is a hyperbola of vertex separation 2a whose fociare the two given points.

14.3 Path followed by a particle

The path followed by any particle in the classical Keplerproblem is a conic section. In particular, if the total en-ergy E of the particle is greater than zero (i.e., if the parti-cle is unbound), the path of such a particle is a hyperbola.This property is useful in studying atomic and sub-atomicforces by scattering high-energy particles; for example,

the Rutherford experiment demonstrated the existence ofan atomic nucleus by examining the scattering of alphaparticles from gold atoms. If the short-range nuclear in-teractions are ignored, the atomic nucleus and the alphaparticle interact only by a repulsive Coulomb force, whichsatisfies the inverse square law requirement for a Keplerproblem.

14.4 Korteweg-de Vries equation

The hyperbolic trig function sech x appears as one solu-tion to the Korteweg-de Vries equation which describesthe motion of a soliton wave in a canal.

14.5 Angle trisection

A B

O

PP'

.

. .. .l

Trisecting an angle (AOB) using a hyperbola of eccentricity 2(yellow curve)

As shown first by Apollonius of Perga, a hyperbola can beused to trisect any angle, a well studied problem of geom-etry. Given an angle, first draw a circle centered at its ver-tex O, which intersects the sides of the angle at points AandB. Next draw the line throughA andB and its perpen-dicular bisector ℓ . Construct a hyperbola of eccentricityε=2 with ℓ as directrix and B as a focus. Let P be the in-tersection (upper) of the hyperbola with the circle. AnglePOB trisects angle AOB. To prove this, reflect the linesegmentOP about the line ℓ obtaining the point P' as theimage of P. SegmentAP' has the same length as segmentBP due to the reflection, while segment PP' has the samelength as segment BP due to the eccentricity of the hy-perbola. As OA, OP', OP and OB are all radii of thesame circle (and so, have the same length), the trianglesOAP', OPP' and OPB are all congruent. Therefore, theangle has been trisected, since 3×POB = AOB.[13]

14 19 EXTERNAL LINKS

14.6 Efficient portfolio frontier

In portfolio theory, the locus of mean-variance efficientportfolios (called the efficient frontier) is the upper halfof the east-opening branch of a hyperbola drawn withthe portfolio return’s standard deviation plotted horizon-tally and its expected value plotted vertically; accordingto this theory, all rational investors would choose a port-folio characterized by some point on this locus.

15 Extensions

The three-dimensional analog of a hyperbola is ahyperboloid. Hyperboloids come in two varieties, thoseof one sheet and those of two sheets. A simple way ofproducing a hyperboloid is to rotate a hyperbola aboutthe axis of its foci or about its symmetry axis perpendic-ular to the first axis; these rotations produce hyperboloidsof two and one sheet, respectively.

16 See also

16.1 Other conic sections

• Circle

• Ellipse

• Parabola

16.2 Other related topics

• Apollonius of Perga, the Greek geometer who gavethe ellipse, parabola, and hyperbola the names bywhich we know them.

• Elliptic coordinates, an orthogonal coordinate sys-tem based on families of ellipses and hyperbolas.

• Hyperbolic function

• Hyperbolic growth

• Hyperbolic partial differential equation

• Hyperbolic sector

• Hyperbolic structure

• Hyperbolic trajectory

• Hyperboloid

• Multilateration

• Rotation of axes

• Translation of axes

• Unit hyperbola

17 Notes[1] Heath, Sir Thomas Little (1896), “Chapter I. The discov-

ery of conic sections. Menaechmus”, Apollonius of Perga:Treatise on Conic Sections with Introductions Including anEssay on Earlier History on the Subject, Cambridge Uni-versity Press, pp. xvii–xxx.

[2] Boyer, Carl B.; Merzbach, Uta C. (2011), A History ofMathematics, Wiley, p. 73, ISBN 9780470630563, Itwas Apollonius (possibly following up a suggestion ofArchimedes) who introduced the names “ellipse” and “hy-perbola” in connection with these curves.

[3] Korn, Granino A. and Korn, Theresa M. MathematicalHandbook for Scientists and Engineers: Definitions, The-orems, and Formulas for Reference and Review, DoverPubl., second edition, 2000: p. 40.

[4] In some literature the value of a is taken negative for ahyperbola (the negative of half the distance between thetwo arms of the hyperbolameasured along themajor axis).This allows some formulas to be applicable to ellipses aswell as to hyperbolas.

[5] Weisstein, Eric W. “Rectangular Hyperbola.” FromMathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RectangularHyperbola.html

[6] Weisstein, Eric W. “Jerabek Hyperbola.”From MathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/JerabekHyperbola.htm

[7]

[8] Spain, Barry. Analytical Conics. Dover Publ., 2007.

[9]

[10] Mitchell, DouglasW., “A property of hyperbolas and theirasymptotes”, Mathematical Gazette 96, July 2012, 299-301.

[11]

[12]

[13] This construction is due to Pappus of Alexandria (circa300 A.D.) and the proof comes from Kazarinoff (1970,pg. 62).

18 References

• Kazarinoff, Nicholas D. (2003), Ruler and theRound, Mineola, N.Y.: Dover, ISBN 0-486-42515-0

19 External links

• Hazewinkel, Michiel, ed. (2001), “Hyperbola”,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

15

• Apollonius’ Derivation of the Hyperbola atConvergence

• Unit hyperbola at PlanetMath.org.

• Conic section at PlanetMath.org.

• Conjugate hyperbola at PlanetMath.org.

• Weisstein, Eric W., “Hyperbola”, MathWorld.

16 20 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

20 Text and image sources, contributors, and licenses

20.1 Text• Hyperbola Source: https://en.wikipedia.org/wiki/Hyperbola?oldid=687870464 Contributors: AxelBoldt, Brion VIBBER, Tarquin, As-troNomer~enwiki, Xaonon, Arvindn, PierreAbbat, Patrick, Michael Hardy, Tongpoo, Wapcaplet, Cyde, Ellywa, Ojs, AugPi, Hemmer,Charles Matthews, Dysprosia, Saltine, Alembert~enwiki, Sabbut, Tjdw, Donarreiskoffer, Fredrik, CHz, Wereon, Jleedev, Tosha, Decrypt3,Giftlite, Jyril, Gene Ward Smith, Harp, Wwoods, Alison, Tom-, Jorge Stolfi, Macrakis, LiDaobing, Alberto da Calvairate~enwiki, Me-likamp, Jossi, Secfan, Maximaximax, Sam Hocevar, Didactohedron, Qef, Archer3, JTN, Discospinster, FiP, Paul August, Brian0918,Rgdboer, Spoon!, Viriditas, Reubot, Alansohn, Mickeyreiss, Wtmitchell, Jheald, Ceyockey, Deror avi, Postrach, Velho, Woohookitty,Linas, Thruston, SeventyThree, Graham87, Magister Mathematicae, Rjwilmsi, HannsEwald, Zhurovai, FlaBot, Mathbot, Alphachimp,Chobot, Krishnavedala, DVdm, Twoeyedhuman, Stephen Compall, YurikBot, Wavelength, Vagodin, Sceptre, Akamad, Rick Norwood,Grafen, Raven4x4x, Cheeser1, BOT-Superzerocool, Cmglee, SmackBot, RDBury, PEHowland, Reedy, Jeppesn, David.Mestel, Unyoyega,AndyZ, Eskimbot, Hmains, Keegan, MalafayaBot, Dabigkid, DHN-bot~enwiki, Colonies Chris, Hengsheng120, Matthijs, SundarBot,RandomP, Acdx, Spiritia, Phancy Physicist, MarkSutton, Mets501, Laurens-af, Tawkerbot2, The Letter J, Vaughan Pratt, Mikiemike,CBM, Kewldude606, Frankshiv, Nnp, Doctormatt, WillowW, MuKinpatsuDijou, Goldencako, Thijs!bot, Wikid77, RatedRestricted,A3RO, Geekdog, Jj137, Phanerozoic, Eleos, JAnDbot, Greensburger, LittleOldMe, Ibapah, Maniwar, Bcherkas, Donnyton, David Epp-stein, Styrofoam1994, MartinBot, Gcranston, Captain panda, SharkD, Krishnachandranvn, Policron, Zojj, 2help, Bcnof, LaughingVul-can, 106er, Gogobera, Deor, VolkovBot, Joeoettinger, AlnoktaBOT, TXiKiBoT, Anonymous Dissident, Melsaran, LeaveSleaves, Induc-tiveload, Deipnosopher, Jesin, Eubulides, Seresin, SieBot, Unamed102, Ceroklis, Gerakibot, MyNameIsHi, Wombatcat, Dhexus~enwiki,Nic bor, Sgtzac1, ClueBot, Plastikspork, Arakunem, AlasdairGreen27, Blanchardb, LizardJr8, DragonBot, Tylerdmace, Cenarium, Qwfp,Deutschnummereins, Crowsnest, DumZiBoT, Noeatingallowed, Addbot, Fgnievinski, Computer Guy 990, Ronhjones, MrOllie, An-naFrance, Kiril Simeonovski, Zorrobot, Luckas-bot, Yobot, Stamcose, THEN WHO WAS PHONE?, AnomieBOT, Götz, To Fight aVandal, Pontificalibus, RibotBOT, Rainald62, Constructive editor, FrescoBot, GuidoB, Pinethicket, Bookerj, RedBot, 1to0to-1, Kevin-tampa5, Jujutacular, Turian, Rausch, Reconsider the static, December21st2012Freak, DixonDBot, Michael9422, Duoduoduo, Garglax,EmausBot, Mz7, Odysseus1479, Tim Zukas, Puffin, 28bot, Anita5192, ClueBot NG, Plantkeeper, Wcherowi, Chester Markel, Frietjes,Joel B. Lewis, Anxiousswift, Hallows AG, Davidiad, Six55eee6, Wing139, Benlcox, Hmainsbot1, Argon34, Ag2gaeh, Bigsexy58, Trixie05,Library Guy, Joshwitges, Loraof, KasparBot, Kafishabbir, Tarang.mungole and Anonymous: 217

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