Hyperbola

Etymology and history

The word "hyperbola" derives from the Greek ὑπερβολή, meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have been coined by Apollonius of Perga (c. 262–c. 190 BC) in his definitive work on the conic sections, the Conics. The names of the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than the segment or exceed the segment.

Definition of a hyperbola as locus of points

A hyperbola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

A hyperbola is a set of points, such that for any point P of the set the absolut difference of the distances | P F 1 | ,   | P F 2 | to two fixed points F 1 , F 2 , the foci, is constant, usually assigned by 2 a ,   a > 0   :

H = { P ∣ | | P F 2 | − | P F 1 | | = 2 a }   .

The midpoint M between the foci is called center of the hyperbola. The line through the foci is called major axis. It contains the vertices V 1 , V 2 , which have distance a to the center. The distance c of the foci to the center is called focal distance or linear eccentricity. The quotient c a is the eccentricity e .

Remark:
The equation | | P F 2 | − | P F 1 | | = 2 a can be considered in a different way (see picture):
If c 2 is the circle with midpoint F 2 and radius 2 a , then the distance of a point P of the right branch to the circle c 2 equals the distance to focus F 1 :

c 2 is called director circle (related to focus F 2 ) of the hyperbola. In order to get the left branch of the hyperbola, one has to use the directrix circle related to F 1 . This property should not be confused with the definition of a hyperbola with help of two directrices (lines) below.

equation

Usually one introduces Cartesian coordinates such that the origin is the center of the hyperbola and the x-axis is the major axis. In this case the hyperbola is called east-west-opening and

the foci are the points F 1 = ( c , 0 ) ,   F 2 = ( − c , 0 ) ,the vertices are V 1 = ( a , 0 ) ,   V 2 = ( − a , 0 ) .

For an arbitrary point ( x , y ) the distance to focus ( c , 0 ) is ( x − c ) 2 + y 2 and to the second focus ( x + c ) 2 + y 2 . Hence the point ( x , y ) is on the hyperbola if the following condition is fulfilled

( x − c ) 2 + y 2 − ( x + c ) 2 + y 2 = ± 2 a   .

Removing the square roots by suitable squarings and with the abbriviation b 2 = c 2 − a 2 one gets at least the equation of the hyperbola:

x 2 a 2 − y 2 b 2 = 1   .

The shape parameters a , b are called semi major axis and semi minor axis or conjugate axis. Against an ellipse a hyperbola has only two vertices: ( a , 0 ) , ( − a , 0 ) . The two points ( 0 , b ) , ( 0 , − b ) on the conjugate axis are not on the hyperbola !

From the equation one recognizes: The hyperbola is symmetric to both the coordinate axes and hence symmetric to the origin, too.

asymptotes

Solving the equation (above) of the hyperbola for y one gets

y = ± b x 2 a 2 − 1 .

Herefrom one recognizes that the hyperbola comes closer to the two lines

y = ± b a x

for large values of | x | . These two lines concide with the center (origin) and are called asymptotes of the hyperbola x 2 a 2 − y 2 b 2 = 1   .

semi latus rectum

The length of the chord through one of the foci, which is perpendicular to the major axis of the hyperbola is called latus rectum. One half of it is the semi latus rectum p . A calculation shows

p = b 2 a .

Further meaning: p is the radius of curvature of the osculating circles at the vertices.

tangent

The simplest way to determine the equation of the tangent at a point ( x 0 , y 0 ) is to differentiate implicitely the equation x 2 a 2 − y 2 b 2 = 1 of the hyperbola. One gets

2 x a 2 − 2 y y ′ b 2 = 0   →   y ′ = x y b 2 a 2   →   y = x 0 y 0 b 2 a 2 ( x − x 0 ) + y 0 .

With respect of x 0 2 a 2 − y 0 2 b 2 = 1 one gets the equation of the tangent at point ( x 0 , y 0 ) :

x 0 a 2 x − y 0 b 2 y = 1.

rectangular hyperbola

In case of a = b the hyperbola is called rectangular (or equilateral), because its asymptotes intersect rectangular. The linear eccentricity is c = 2 a , the eccentricity e = 2 and semi latus rectum p = a .

parametric representation with hyperbolic sine/cosine

Using the hyperbolic sine and cosine functions cosh , sinh one gets a parametric representation of the hyperbola x 2 a 2 − y 2 b 2 = 1 , which is similar to the parametric representation of an ellipse:

( ± a cosh ⁡ t , b sinh ⁡ t ) , t ∈ R   .

(It is cosh 2 ⁡ t − sinh 2 ⁡ t = 1   ! )

conjugate hyperbola

If one exchanges x and y , one gets the equation of the conjugate hyperbola (s. picture):

y 2 a 2 − x 2 b 2 = 1   , or written as

x 2 a 2 − y 2 b 2 = − 1   .

Hyperbola with equation y=A/x

If the xy-coordinate system is rotated about the origin by the angle − 45 ∘ and new coordinates ξ , η are assigned, then x = ξ + η 2 , y = − ξ + η 2 .
The rectangular hyperbola x 2 − y 2 a 2 = 1 (whose semi axes are equal) has the new equation 2 ξ η a 2 = 1 . Solving for η yields η = a 2 / 2 ξ   .

Thus, in an xy-coordinate system the graph of a function f : x ↦ A x , A > 0 , with equation

y = A x , A > 0 , is a rectangular hyperbola with

the coordinate axes as asymptotes,

the line y = x as major axis ,

the center ( 0 , 0 ) and the semi-axis a = b = 2 A ,

the vertices ( A , A ) , ( − A , − A ) ,

the semi latus rectum and radius of curvature at the vertices p = a = 2 A ,

the linear eccentricity e = 2 A and the eccentricity e = 2 ,

the tangent y = − A x 0 2 x + 2 A x 0 at point ( x 0 , A / x 0 ) .

A rotation of the hyperbola by + 45 ∘ results in a rectangular hyperbola with equation

y = − A x , A > 0 ,

the semi axes a = b = 2 A ,

the line y = − x as major axis ,

the vertices ( − A , A ) , ( A , − A ) .

Shifting the hyperbola with equation y = A x ,   A ≠ 0   , so that the new center is ( c 0 , d 0 ) , yields the new equation

y = A x − c 0 + d 0 ,

and the new asymptotes are x = c 0 and y = d 0 .
The shape parameters a , b , p , c , e remain unchanged.

Definition of a hyperbola by the directrix property

The two parallel lines with distance d = a 2 c to the minor axis are called directrices of the hyperbola (see picture). The following statement is true:

For an arbitrary point P of the hyperbola the quotient of the distances to one focus and to the corresponding directrix (see picture) is equal to the eccentricity:

The proof for the pair F 1 , l 1 follows from the fact that | P F 1 | 2 = ( x − c ) 2 + y 2 ,   | P l 1 | 2 = ( x − a 2 c ) 2 and y 2 = b 2 a 2 x 2 − b 2 fulfill the equation

| P F 1 | 2 − c 2 a 2 | P l 1 | 2 = 0   .

Analogously the second case is proven.

The inverse statement is true either and can be used to define a hyperbola (similar to a parabola):

For any point F (focus), any line l (directrix) not through F and any real number e with e > 1 the set of points (locus of points), for which the quotient of the distances to the point and to the line is e

is a hyperbola.

(The choice e = 1 yields a parabola and if e < 1 one gets an ellipse.)

proof

Let be F = ( f , 0 ) ,   e > 0 and assume ( 0 , 0 ) is a point of the curve. Hence the directrix l has equation x = − f e . For P = ( x , y ) one gets from | P F | 2 = e 2 | P l | 2 the equations

( x − f ) 2 + y 2 = e 2 ( x + f e ) 2 = ( e x + f ) 2 and herefrom x 2 ( e 2 − 1 ) + 2 x f ( 1 + e ) − y 2 = 0.

The abbriviation p = f ( 1 + ε ) yields

x 2 ( e 2 − 1 ) + 2 p x − y 2 = 0.

This is the equation of an ellipse ( e < 1 ) or a parabola ( e = 1 ) or a hyperbola ( e > 1 ). All these non degenerate conics have the origin as vertex in common (see picture).

If e > 1 one introduces new parameters a , b such that e 2 − 1 = b 2 a 2 ,   p = b 2 a the equation above turns into

( x + a ) 2 a 2 − y 2 b 2 = 1   ,

which is the equation of a hyperbola with center ( − a , 0 ) , the x-axis as major axis and the major/minor semi axis a , b .

Hyperbola as plane section of a cone

The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see picture: red curve). In order to proof the defining property of a hyperbola (see above) one uses two Dandelin spheres d 1 , d 2 , which are spheres who touches the cone along circles c 1 , c 2 and the intersecting (hyperbola) plane at points F 1 and F 2 . It turns out: F 1 , F 2 are the foci of the hyperbola.

1. Let be P an arbitrary point of the intersection curve .
2. The generator (line) of the cone containing P intersects circle c 1 at point A and circle c 2 at a point B .
3. The linesegments P F 1 ¯ and P A ¯ are tangential to the sphere d 1 and hence of equal length.
4. The line segments P F 2 ¯ and P B ¯ are tangential to the sphere d 2 and hence of equal length.
5. The result is: | P F 1 | − | P F 2 | = | P A | − | P B | = | A B | is independent of the hyperbola point P .

The tangent bisects the lines to the foci

For a hyperbola the following statement is true:

The tangent at a point P bisects the lines P F 1 ¯ , P F 2 ¯ .

proof

Let be L the point on the line P F 2 ¯ with the distance 2 a to the focus F 2 (see picture, a is the semi major axis of the hyperbola). Line w is the bisector of the lines P F 1 ¯ , P F 2 ¯ . In order to prove that w is the tangent line at point P , one checks that any point Q on line w which is different to P can not be on the hyperbola. Hence w has with the hyperbola only point P in common and is therefor the tangent at point P .
From the picture and the triangle inequality one recognizes that | Q F 2 | < | L F 2 | + | Q L | = 2 a + | Q F 1 | holds, that means: | Q F 2 | − | Q F 1 | < 2 a . But if Q would be a point of the hyperbola the difference should be 2 a .

Midpoints of parallel chords

The midpoints of parallel chords of a hyperbola lie on a line through the center (see pictutre).

The points of any chord may lie on different branches of the hyperbola.

The proof of the property on midpoints is done at its best for the hyperbola y = 1 / x . Because any hyperbola is an affine image of the hyperbola y = 1 / x (see section below) and an affine transformation respects parallelism and midpoints of line segments, the property is true for all hyperbolas:
For two points P = ( x 1 , 1 x 1 ) ,   Q = ( x 2 , 1 x 2 ) of the hyperbola y = 1 / x

the midpoint of the chord is M = ( x 1 + x 2 2 , ⋯ ) = ⋯ = x 1 + x 2 2 ( 1 , 1 x 1 x 2 )   , the slope of the chord is 1 x 2 − 1 x 1 x 2 − x 1 = ⋯ = − 1 x 1 x 2   .

For parallel chords the slope is constant and the midpoints of the parallel chords on the line y = 1 x 1 x 2 x   .

Consequence: For any pair of points P , Q of a chord there exists a skew reflection with an axis (set of fixed points) passing the center of the hyperbola, which exchanges the points P , Q and leaves the hyperbola (as whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line m , where all point-image pairs are on a line perpendicular to m .

Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed either. Hence the midpoint M of a chord P Q divides the related line segment P ¯ Q ¯ between the asymptotes into halves, too. That means, that | P P ¯ | = | Q Q ¯ | . This property can be used for the construction of further points Q of the hyperbola if a point P and the asymptotes are given.

If the chord degenerates into a tangent, then the touching point divides the line segment between the asymptotes in two halves.

Steiner generation of a hyperbola

The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section:

Given two pencils B ( U ) , B ( V ) of lines at two points U , V (all lines containing U and V resp.) and a projective but not perspective mapping π of B ( U ) onto B ( V ) . Then the intersection points of corresponding lines form a non-degenerate projective conic section.

For the generation of points of the hyperbola x 2 a 2 − y 2 b 2 = 1 one uses the pencils at the vertices V 1 , V 2 . Let be P = ( x 0 , y 0 ) a point of the hyperbola and A = ( a , y 0 ) , B = ( x 0 , 0 ) . The line segment B P ¯ is divided into n equal spaced segments and this division is projected parallely with the diagonal A B as direction onto the line segment A P ¯ (see picture). The parallel projection is part of the projective mapping between the pencils at V 1 and V 2 needed. The intersection points of any two related lines S 1 A i und S 2 B i are points of the uniquely defined hyperbola.

Remark: The subdivision could be extended beyond the points A and B in order get more points. But the determination of the intersection points would become more inaccurate. So, the better idea is: extending the points already constructed by symmetry (see animation).

Remark:

1. The Steiner generation exists for ellipses and parabolas, too.
2. The Steiner generation is sometimes called parallelogram method because one can use other points than the vertices which starts with a parallelogram instead of a rectangle.

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

e = B C ¯ 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) in the plane,

A x x x 2 + 2 A x y x y + A y y y 2 + 2 B x x + 2 B y y + C = 0 ,

provided that the constants A_xx, A_xy, A_yy, B_x, B_y, and C satisfy the determinant condition

D := | A x x A x y A x y A y y | < 0.

This determinant is conventionally called the discriminant of the conic section.

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

Δ := | A x x A x y B x A x y A y y B y B x B y C | = 0.

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

Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in Conic section#Eccentricity in terms of parameters of the quadratic form.

The center (x_c, y_c) of the hyperbola may be determined from the formulae

x c = − 1 D | B x A x y B y A y y | ; y c = − 1 D | A x x B x A x y B y | .

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

A x x ξ 2 + 2 A x y ξ η + A y y η 2 + Δ D = 0.

The principal axes of the hyperbola make an angle φ with the positive x-axis that is given by

tan ⁡ 2 φ = 2 A x y A x x − A y y .

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

x 2 a 2 − y 2 b 2 = 1.

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

a 2 = − Δ λ 1 D = − Δ λ 1 2 λ 2 , b 2 = − Δ λ 2 D = − Δ λ 1 λ 2 2 ,

where λ₁ and λ₂ are the roots of the quadratic equation

λ 2 − ( A x x + A y y ) λ + D = 0.

For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is

x 2 a 2 − 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 by

E = A x x x 0 + A x y y 0 + B x , F = A x y x 0 + A y y 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 ( x − x 0 ) − E ( y − y 0 ) = 0.

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

From the equation

x 2 a 2 − y 2 b 2 = 1 , 0 < b ≤ a ,

the left focus is ( − a e , 0 ) and the right focus is ( a e , 0 ) , where e is the eccentricity. Denote the distances from a point (x, y) to the left and right foci as r 1 and r 2 . For a point on the right branch,

r 1 − r 2 = 2 a ,

and for a point on the left branch,

r 2 − r 1 = 2 a .

This can be proved as follows:

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

r 1 2 = ( x + a e ) 2 + y 2 = x 2 + 2 x a e + a 2 e 2 + ( x 2 − a 2 ) ( e 2 − 1 ) = ( e x + a ) 2 .

To the right focal point the distance is

r 2 2 = ( x − a e ) 2 + y 2 = x 2 − 2 x a e + a 2 e 2 + ( x 2 − a 2 ) ( e 2 − 1 ) = ( e x − a ) 2 .

If (x,y) is a point on the right branch of the hyperbola then e x > a and

r 1 = e x + a , r 2 = e x − a .

Subtracting these equations one gets

r 1 − r 2 = 2 a .

If (x,y) is a point on the left branch of the hyperbola then e x < − a and

r 1 = − e x − a , r 2 = − e x + a .

Subtracting these equations one gets

r 2 − r 1 = 2 a .

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

x 2 a 2 − y 2 b 2 = 1 ,

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

r = − e x − a .

Introducing polar coordinates ( r   ,   θ ) with origin at the left focal point, the coordinates relative to the canonical coordinate system are

x   =   − a e + r cos ⁡ θ , y   = r sin ⁡ θ ,

and the equation above takes the form

r = − e ( − a e + r cos ⁡ θ ) − a

from which it follows that

r = a ( e 2 − 1 ) 1 + e cos ⁡ θ .

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

The polar angle θ of a point on a hyperbola relative to the near focal point as described above is called the true anomaly of the point.

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 plane. 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 shown in Figure 10. 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.

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.

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 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.

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

r = a ( e 2 − 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 coordinate system"

x = R cos ⁡ t y = R sin ⁡ t

one has that

R 2 = b 2 e 2 cos 2 ⁡ t − 1 .

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

− arccos ⁡ ( 1 e ) < t < arccos ⁡ ( 1 e ) .

Parametric equations

East-west opening hyperbola:

x = a sec ⁡ t + h y = b tan ⁡ t + k or x = ± a cosh ⁡ t + h y = b sinh ⁡ t + k

North-south opening hyperbola:

x = b tan ⁡ t + h y = a sec ⁡ t + k or x = b sinh ⁡ t + h y = ± a cosh ⁡ t + k

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 coordinates in two dimensions. These hyperbolas 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 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.

Hyperbola as an affine image of the unit hyperbola x²-y²=1

Another definition of a hyperbola uses affine transformations:

Any hyperbola is the affine image of the unit hyperbola with equation x 2 − y 2 = 1 .

An affine transformation of the Euclidean plane has the form x → → f → 0 + A x → , where A is a regular matrix (its determinant is not 0) and f → 0 is an arbitrary vector. If f → 1 , f → 2 are the column vectors of the matrix A , the unit hyperbola ( ± cosh ⁡ ( t ) , sinh ⁡ ( t ) ) , t ∈ R , is mapped onto the hyperbola

x → = p → ( t ) = f → 0 ± f → 1 cosh ⁡ t + f → 2 sinh ⁡ t   .

f → 0 is the center, f → 0 + f → 1 a point of the hyperbola and f → 2 a tangent vector at this point. In general the vectors f → 1 , f → 2 are not perpendicular. That means, in general f → 0 ± f → 1 are not the vertices of the hyperbola. But f → 1 ± f → 2 point into the directions of the asymptotes.

The tangent vector at point p → ( t ) is

p → ′ ( t ) = f → 1 sinh ⁡ t + f → 2 cosh ⁡ t   .

Because at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter t 0 of a vertex from the equation

p → ′ ( t ) ⋅ ( p → ( t ) − f → 0 ) = ( f → 1 sinh ⁡ t + f → 2 cosh ⁡ t ) ⋅ ( f → 1 cosh ⁡ t + f → 2 sinh ⁡ t ) = 0

and hence from

coth ⁡ ( 2 t 0 ) = − f → 1 2 + f → 2 2 2 f → 1 ⋅ f → 2   ,

which yields

t 0 = 1 4 ln ⁡ ( f → 1 − f → 2 ) 2 ( f → 1 + f → 2 ) 2 .

(The formulae cosh 2 ⁡ x + sinh 2 ⁡ x = cosh ⁡ 2 x ,   2 sinh ⁡ x cosh ⁡ x = sinh ⁡ 2 x ,   arcoth ⁡ x = 1 2 ln ⁡ x + 1 x − 1 were used.)

The two vertices of the hyperbola are f → 0 ± ( f → 1 cosh ⁡ t 0 + f → 2 sinh ⁡ t 0 ) .

The advantage of this definition is that one gets a simple parametric representation of an arbitrary hyperbola, even in the space, if the vectors f → 0 , f → 1 , f → 2 are vectors of the Euclidean space.

Hyperbola as an affine image of the hyperbola y=1/x

Because the unit hyperbola x 2 − y 2 = 1 is affinely equivalent to the hyperbola y = 1 / x , an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola y = 1 / x   :

x → = p → ( t ) = f → 0 + f → 1 t + f → 2 1 t , t ≠ 0   .

M : f → 0 is the center of the hyperbola, the vectors f → 1 , f → 2 have the directions of the asymptotes and f → 1 + f → 2 is a point of the hyperbola. The tangent vector is

p → ′ ( t ) = f → 1 − f → 2 1 t 2 .

At a vertex the tangent is perpendicular to the major axis. Hence

p → ′ ( t ) ⋅ ( p → ( t ) − f → 0 ) = ( f → 1 − f → 2 1 t 2 ) ⋅ ( f → 1 t + f → 2 1 t ) = f → 1 2 t − f → 2 2 1 t 3 = 0.

and the parameter of a vertex is

t 0 = ± f → 2 2 f → 1 2 4 .

For | f → 1 | = | f → 2 | one gets t 0 = ± 1 and f → 0 ± ( f → 1 + f → 2 ) are the verteces of the hyperbola.

The following properties of a hyperbola are easily proven using the representation of a hyperbola introduced in this section.

Tangent construction

The tangent vector can be rewritten by factorization:

p → ′ ( t ) = 1 t ( f → 1 t − f → 2 1 t )   .

This means that

the diagonal A B of the parallelogram M :   f → 0 ,   A = f → 0 + f → 1 t ,   B :   f → 0 + f → 2 1 t ,   P :   f → 0 + f → 1 t + f → 2 1 t is parallel to the tangent at the hyperbola point P (see picture).

This property provides a way to construct the tangent at a point on the hyperbola.

This property of a hyperbola is an affine version of the 3-point-degeneration of Pascal's theorem.

Point construction

For a hyperbola with parametric representation x → = p → ( t ) = f → 1 t + f → 2 1 t (for simplicity the center is the origin) the following is true:

For any two points P 1 :   f → 1 t 1 + f → 2 1 t 1 ,   P 2 :   f → 1 t 2 + f → 2 1 t 2 the points

A :   a → = f → 1 t 1 + f → 2 1 t 2 ,   B :   b → = f → 1 t 2 + f → 2 1 t 1 are collinear with the center of the hyperbola (see picture).

The simple proof is a consequence of the equation 1 t 1 a → = 1 t 2 b → .

This property provides a possibilty to construct points of a hyperbola if the asymptotes and one point are given.

This property of a hyperbola is an affine version of the 4-point-degeneration of Pascal's theorem.

Tangent-asymptotes-triangle

For simplicity the center of the hyperbola may be the origin and the vectors f → 1 , f → 2 have equal length. If the last asumption is not fulfilled one can first apply a parameter transformation (see above) in order to make the assumption true. Hence ± ( f → 1 + f → 2 ) are the vertices, ± ( f → 1 − f → 2 ) span the minor axis and one gets | f → 1 + f → 2 | = a and | f → 1 − f → 2 | = b .

For the intersection points of the tangent at point p → ( t 0 ) = f → 1 t 0 + f → 2 1 t 0 with the asymptotes one gets the points

C = 2 t 0 f → 1 ,   D = 2 t 0 f → 2 .

The area of the triangle M , C , D can be calculated by a 2x2-determinant:

A = 1 2 | det ( 2 t 0 f → 1 , 2 t 0 f → 2 ) | = 2 | det ( f → 1 , f → 2 ) |

(see rules for determinants). | det ( f → 1 , f → 2 ) | is the area of the rhombus generated by f → 1 , f → 2 . The area of a rhombus is equal to one half of the product of its diagonals. The diagonals are the semi-axes a , b of the hyperbola. Hence:

The area of the triangle M , C , D is independent of the point of the hyperbola: A = a b .

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 concurrent: (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 on the hyperbola 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. This product is a 2 + b 2 4 , where a and b are the semi-major and semi-minor axes respectively.

The product of the distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes, along lines perpendicular to the asymptotes, is a constant independent of the location of P: specifically, a 2 b 2 a 2 + b 2 , which also equals ( b / e ) 2 , where e is the eccentricity of one of the hyperbolas and a and b are its semi-major and semi-minor axes respectively.

The product of the slopes of lines from a point on the east-west opening hyperbola x 2 a 2 − y 2 b 2 = 1 to the two vertices is b 2 / a 2 , 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 theorem on the right triangle with these two segments as legs shows that a 2 + b 2 = c 2 , where c is the semi-focal length (the distance from a focus to the hyperbola's center).

Sundials

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 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 (this path is called the declination line). 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.

Multilateration

A hyperbola is the basis for solving multilateration 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.

Path followed by a particle

The path 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 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.

Korteweg–de Vries equation

The hyperbolic trig function sech x appears as one solution to the Korteweg–de Vries equation which describes the motion of a soliton wave in a canal.

Angle trisection

As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, a well studied problem of geometry. Given an angle, first draw a circle centered at its vertex O, which intersects the sides of the angle at points A and B. Next draw the line segment with endpoints A and B and its perpendicular bisector ℓ . Construct a hyperbola of eccentricity e=2 with ℓ as directrix and B as a focus. Let P be the intersection (upper) of the hyperbola with the circle. Angle POB trisects angle AOB. To prove this, reflect the line segment OP about the line ℓ obtaining the point P' as the image of P. Segment AP' has the same length as segment BP due to the reflection, while segment PP' has the same length as segment BP due to the eccentricity of the hyperbola. As OA, OP', OP and OB are all radii of the same circle (and so, have the same length), the triangles OAP', OPP' and OPB are all congruent. Therefore, the angle has been trisected, since 3×POB = AOB.

Efficient portfolio frontier

In portfolio theory, the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.

Extensions

The three-dimensional analog of a hyperbola is a hyperboloid. Hyperboloids 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.