Hyperbola

A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.
Hyperbolae in the physical world: three cones of light of different widths and intensities are generated by a (roughly) downwards-pointing halogen lamp and its holder. Each cone of light is drawing a branch of a hyperbola on a nearby vertical wall.
Hyperbolas produced by interference of waves

In mathematics a hyperbola is a type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror 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 cone. The other conic sections are the parabola, the ellipse, and the circle (the circle is a special case of the ellipse). Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a straight line on the surface of the cone makes with the axis of the cone. If the angle between the plane and the axis is less than the angle between the line on the cone and the axis, or if the plane is parallel to the axis, then the plane intersects both halves of the double cone and the conic is a hyperbola.

Hyperbolas arise in practice in many ways: as the curve representing the function $f(x) = 1/x$ in the Cartesian plane, as the appearance of a circle viewed from within it, as the path followed by the shadow of the tip of a sundial, as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition comet (one travelling too fast to ever return to the solar system), as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same), and so on.

Each branch of the hyperbola consists of two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms one from each branch tend in the limit to a common line, called the asymptote of those two arms. There are therefore two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve $f(x) = 1/x$ the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry used in both relativity and quantum mechanics which is not Euclidean).

History

The word "hyperbola" derives from the Greek ὑπερβολή, meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. The term hyperbola is believed to have been coined by Apollonius of Perga (c. 262–c. 190 BC)[citation needed] in his definitive work on the conic sections, the Conics. For comparison, the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "comparable"; these terms may refer to the eccentricity of these curves, which is greater than one (hyperbola), less than one (ellipse) and exactly one (parabola).

Nomenclature and features

The hyperbola consists of the red curves. The asymptotes of the hyperbola are shown as blue dashed lines and intersect at the center 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 conjugate axis. The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2. The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green). The two vertices are located on the transverse axis at ±a relative to the center. So the parameters are: a — distance from center C to either vertex
b — length of a perpendicular segment from each vertex to the asymptotes
c — distance from center C to either Focus point, F1 and F2, and
θ — angle formed by each asymptote with the transverse axis.

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 consists of two disconnected curves called its arms or branches.

The points on the two branches that are closest to each other are called the vertices; they are the points where the curve has its smallest radius of curvature. The line segment connecting the vertices is called the transverse axis or major axis, corresponding to the major diameter of an ellipse. The midpoint of the transverse axis is known as the hyperbola's center. The distance a from the center to each vertex is called the semi-major axis. Outside of the transverse axis but on the same line are the two focal points (foci) of the hyperbola. The line through these five points is one of the two principal axes of the hyperbola, the other being the perpendicular bisector of the transverse axis. The hyperbola has mirror symmetry about its principal axes, and is also symmetric under a 180° turn about its center.

At large distances from the center, the hyperbola approaches two lines, its 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, if the transverse axis is aligned with the x-axis of a Cartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±ba, where b=a×tan(θ) and where θ is the angle between the transverse axis and either asymptote. The distance b (not shown) is the length of the perpendicular segment from either vertex to the asymptotes.

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

If 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 points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides 2a = 2b.

If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as

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

A hyperbola aligned in this way is called an "East-West opening hyperbola". Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola" and has equation

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

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 to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated (rigidly moved in the plane) so that it is centered at the origin. For convenience, hyperbolas are usually analyzed in terms of their centered East-West opening form.

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

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

$\varepsilon = \frac{c}{a} = \frac{\sqrt{a^{2} + b^{2}}}{a} = \sqrt{1 + \frac{b^{2}}{a^{2}}} = \sec \theta .$

For example, the eccentricity of a rectangular hyperbola (θ = 45°, a = b) equals the square root of two: ε =  $\sqrt{2}$.

Every hyperbola has a conjugate hyperbola, in which the transverse and conjugate axes are exchanged without changing the asymptotes. The equation of the conjugate hyperbola of $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = -1$. If the graph of the conjugate hyperbola is rotated 90° to restore the east-west opening orientation (so that x becomes y and vice versa), the equation of the resulting rotated conjugate hyperbola is the same as the equation of the original hyperbola except with a and b exchanged. For example, the angle θ of the conjugate hyperbola equals 90° minus the angle of the original hyperbola. Thus, the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities unless θ = 45° (a rectangular hyperbola). Hence, the conjugate hyperbola does not in general correspond to a 90° rotation of the original hyperbola; the two hyperbolas are generally different in shape.

A few other lengths are used to describe hyperbolas. Consider a line perpendicular to the transverse axis (i.e., parallel to the conjugate axis) that passes through one of the hyperbola's foci. The line segment connecting the two intersection points of this line with the hyperbola is known as the latus rectum and has a length $\frac{2b^{2}}{a}$. The semi-latus rectum l is half of this length, i.e., $l=\frac{b^{2}}{a}$. The focal parameter p is the distance from a focus to its corresponding directrix, and equals $p=\frac{b^{2}}{c}$.

Mathematical definitions

A hyperbola can be defined mathematically in several equivalent ways.

Conic section

Three major types of conic sections.

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 passes through the central apex of the double cone a degenerate hyperbola results — two straight lines that cross at the apex point.

Difference of distances to foci

A hyperbola may be defined equivalently as the locus of points where the absolute value of the difference of the distances to the two 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.

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

Directrix and focus

A hyperbola can be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant $\epsilon$ that is larger than 1. This constant is the eccentricity of the hyperbola. By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex.

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.

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.[1]

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 (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 = xxc and η = yyc, the defining equation of the hyperbola can be written

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

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

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

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

From the equation

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \qquad 0 < b \leq a$

the basic property that with $r_1 \,\!$ and $r_2 \,\!$ being the distances from a point $(x,y) \,\!$ to the left focus $(-a e , 0) \,\!$ and the right focus $(a e , 0) \,\!$ one has for a point on the right branch that

$r_1 - r_2 =2 a\,\!$

and for a point on the left branch that

$r_2 - r_1 =2 a\,\!$

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

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

In the section above it is shown that using the coordinate system in which the equation of the hyperbola takes its canonical form

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

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 $( r\ ,\ \theta)$ with origin at the left focal point the coordinates relative the canonical coordinate system are

$x\ =\ -ae+r \cos \theta$
$y\ =r \sin \theta$

and the equation above takes the form

$r = -e (-ae+r \cos \theta) - a\,\!$

from which follows that

$r = \frac{a(e^2-1)}{1+e\cos \theta}$

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

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

Hyperbola construction using the parallelogram method

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, L1L2 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.

A fourth construction is using the parallelogram method. It is similar to such method for parabola and ellipse construction: certain equally spaced points lying on parallel lines are connected with each other by two straight lines and their intersection point lies on the hyperbola.

Reflections and tangent lines

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

The points $( a\ \cosh\ \mu_k \ ,\ b\ \sinh\ \mu_k)$ with $\mu_k\ =\ 0.3\ k$ for $k=-5,-4, \cdots ,5$

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.

As

$\cosh^2 \mu - \sinh^2 \mu= 1$

one has for any value of $\mu$ that the point

$x = a\ \cosh\ \mu$
$y = b\ \sinh\ \mu$

satisfies the equation

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

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

When μ varies over the interval $-\infty < \mu < \infty$ one gets with this formula all points $(x\ ,\ y)$ on the right branch of the hyperbola.

The left branch for which $x < 0$ is in the same way obtained as

$x = -a\ \cosh\ \mu$
$y = b\ \sinh\ \mu$

In the figure the points $(x_k\ ,\ y_k)$ given by

$x_k = -a\ \cosh \mu _k$
$y_k = b\ \sinh \mu _k$

for

$\mu_k\ =\ 0.3\ k \quad k=-5,-4, \cdots ,5$

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

Relation to other conic sections

There are three major types of conic sections: hyperbolas, 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 canonical equation for a hyperbola is

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

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

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

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

in which the semi-minor axis length b is imaginary. That is, if in the ellipse equation b is replaced by ib where b is real, one obtains the hyperbola equation.

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

$\cosh \mu = \cos i\mu$

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 arc length of a segment of an ellipse can be determined using an incomplete elliptic integral of the second kind. The corresponding arclength of a hyperbola is given by the same function with imaginary parameters b and μ, namely, ib E(iμ, c).

Conic section analysis of the hyperbolic appearance of circles

Figure 2: The hyperbola as a circle on the ground seen in perspective while gazing down slightly, showing circle's tangents as asymptotes. The portion above the horizon is normally invisible.

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 the gaze continues to rise the asymptotes of the hyperbola, if realized concretely, appear coming in from left and right, swinging towards each other and converging at the horizon when the gaze is horizontal. Further elevation of the gaze into the sky then brings the point of convergence of the asymptotes towards the viewer.

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

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

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

A common misperception about the hyperbola is that it is a mathematical curve rarely if ever encountered in daily life. The reality is that one sees a hyperbola whenever catching sight of portion of a circle cut by one's lens plane (and a parabola when the lens plane is tangent to, i.e. just touches, the circle). The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas such as y = 1/x where both branches are on display simultaneously.

Derived curves

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

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.

Coordinate systems

Cartesian coordinates

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

$\frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1.$

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),[2] and b is the semi-minor axis (half the distance between the asymptotes along a line tangent to the hyperbola at a vertex).

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the sides tangent to the hyperbola are 2b in length while the sides that run parallel to the line between the foci (the major axis) are 2a in length. Note that b may be larger than a despite the names minor and major.

If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always 2a.

The eccentricity is given by

$\varepsilon = \sqrt{1+\frac{b^2}{a^2}} = \sec\left(\arctan\left(\frac{b}{a}\right)\right) = \cosh\left(\operatorname{arsinh}\left(\frac{b}{a}\right)\right)$

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

$\left(h\pm c, k\right)$

and for a north-south opening hyperbola are given by

$\left( h, k\pm c\right)$.

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

$x = h\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)$

and for a north-south opening hyperbola are given by

$y = k\pm a \; \cos\left(\arctan\left(\frac{b}{a}\right)\right)$.

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

$r = \frac{a(e^2-1)}{1+e\cos \theta}$

and the range of the true anomaly $\theta$ is:

$-\arccos {\left(-\frac{1}{e}\right)} < \theta < \arccos {\left(-\frac{1}{e}\right)}$

With polar coordinate relative to the "canonical coordinate system"

$x = R\, \cos t$
$y = R\, \sin t$

one has that

$R^2 =\frac{b^2}{e^2 \cos^2 t -1} \,$

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

$-\arccos {\left(\frac{1}{e}\right)} < t < \arccos {\left(\frac{1}{e}\right)}$

Parametric equations

East-west opening hyperbola:

$\begin{matrix} x = a\sec t + h \\ y = b\tan t + k \\ \end{matrix} \qquad \mathrm{or} \qquad\begin{matrix} x = \pm a\cosh t + h \\ y = b\sinh t + k \\ \end{matrix}$

North-south opening hyperbola:

$\begin{matrix} x = b\tan t + h \\ y = a\sec t + k \\ \end{matrix} \qquad \mathrm{or} \qquad\begin{matrix} x = b\sinh t + h \\ y = \pm a\cosh t + k \\ \end{matrix}$

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.

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

$\left(\frac{x}{c \cos\theta}\right)^2 - \left(\frac{y}{c \sin\theta}\right)^2 = 1$

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.

Rectangular hyperbola with horizontal/vertical asymptotes (Cartesian coordinates)

A graph of the rectangular hyperbola $y=\tfrac{1}{x}$, the reciprocal function

Rectangular hyperbolas with the coordinate axes parallel to their asymptotes have the equation

$(x-h)(y-k) = m \, \, \,$.

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 rectangular hyperbolas occurs when the center (h, k) is at the origin:

$y=\frac{m}{x}\,$

describing quantities x and y that are inversely proportional. By rotating the coordinate axes counterclockwise by 45 degrees, with the new coordinate axes labelled $(x',y')$ the equation of the hyperbola is given by canonical form

$\frac{(x')^2}{(\sqrt{2m})^2}-\frac{(y')^2}{(\sqrt{2m})^2}=1$.

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.[3][4]:p.49,ex.7
• 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.[3][5]
• 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.[5]
• The product of the distances from a point P to one of the asymptotes along a line parallel to the other asymptote, and to the second asymptote along a line parallel to the first asymptote, is independent of the location of point P on the hyperbola.[5] If the hyperbola is written in canonical form $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$ then this product is $\frac{a^2+b^2}{4}$.
• The product of the perpendicular distances from a point P on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$ or on its conjugate hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2}=-1$ to the asymptotes is a constant independent of the location of P: specifically, $\frac{a^2b^2}{a^2+b^2}$, which also equals $(b/e)^2$ where e is the eccentricity of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$.[6]
• The product of the slopes of lines from a point on the hyperbola to the two vertices is independent of the location of the point.[7]
• A line segment between the two asymptotes and tangent to the hyperbola is bisected by the tangency point.[4]:p.49,ex.6[7][8]
• 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.[4]:p.49,ex.6 Specifically, the area is ab, where a is the semi-major axis and b is the semi-minor axis.[8]
• 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.[3] 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).

Applications

Hyperbolas as declination lines on a sundial

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.

Trilateration

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.

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 $\operatorname{sech}\, x$ appears as one solution to the Korteweg-de Vries equation which describes the motion of a soliton wave in a canal.

Angle trisection

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

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 through A and B and its perpendicular bisector $\ell$. Construct a hyperbola of eccentricity ε=2 with $\ell$ 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 $\ell$ 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.[9]

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.

Notes

1. ^ Korn, Granino A. and Korn, Theresa M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Dover Publ., second edition, 2000: p. 40.
2. ^ In some literature the value of a is taken negative for a hyperbola (the negative of half the distance between the two arms of the hyperbola measured along the major axis). This allows some formulas to be applicable to ellipses as well as to hyperbolas.
3. ^ a b c [1]
4. ^ a b c Spain, Barry. Analytical Conics. Dover Publ., 2007.
5. ^ a b c [2]
6. ^ Mitchell, Douglas W., "A property of hyperbolas and their asymptotes", Mathematical Gazette 96, July 2012, 299-301.
7. ^ a b [3]
8. ^ a b [4]
9. ^ This construction is due to Pappus of Alexandria (circa 300 A.D.) and the proof comes from Kazarinoff (1970, pg. 62).

References

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