Hyperbola

The least known and appreciated of the conic sections has some interesting applications nevertheless


Definition of the Hyperbola

The hyperbola is the least known and used of the conic sections. We seldom see a hyperbola in daily life, and it seldom appears in decoration or design. In spite of this, it has interesting properties and important applications. There is a literary term, hyperbole, that is the same word in Greek, meaning an excess. How the hyperbola acquired this name is related in Parabola, together with some general information on conic sections, and the focal definition of the hyperbola.

The feature of the hyperbola is its asymptotes. A curve is said to approach a straight line as an asymptote when for any distance ε you may choose, there is always a point on the line beyond which the curve is closer to it than ε. This is, of course, in a certain direction along the line that extends to infinity. A hyperbola has two asymptotes that make equal angles with the coordinate axes and pass through the origin O. Near the origin, the hyperbola passes from one asymptote to the other in a smooth curve. There are two branches of the hyperbola, starting from opposite ends of the asymptotes. For most practical purposes, the hyperbola can be considered as the asymptote itself except in the neighborhood of the origin.

A hyperbola is sketched at the right. The origin is O, and the asyptotes form a symmetrical cross as shown. V and V' are the vertices of the hyperbola, at a distance a on each side of the origin. Perpendicular lines from V and V' define a rectangle by their points of intersection with the asymptotes, and the sides of this rectangle are a and b. Two parameters are required to specify a hyperbola, as for an ellipse. The slope of the asymptotes is |b/a|. Then, the hyperbola can be represented as the quadratic curve (x/a)2 - (y/b)2 = 1, the canonical equation of a hyperbola.

The foci F and F' are located a distance c > a from the origin, where c is the hypotenuse of the right triangle whose sides are a and b. If you draw the reference rectangle for the hyperbola, the foci can be located quite simply by swinging an arc. The difference in the distances F'P and FP from the foci to any point P on the hyperbola is equal to 2a. It is not difficult to prove that this definition is equivalent to the canonical equation. Moreover, as the sketch indicates, the angle between FP and the normal to the hyperbola is equal to the angle between the normal and F'P, so a ray from F is reflected by the hyperbola so that it appears to be coming from the other focus. This is the analogue to the reflecting properties of the parabola and ellipse. The ratio c/a is the eccentricity of the hyperbola, and is > 1. We see that b = a(e2 - 1)1/2, and that the semi-latus rectum p = b2/a. The latter is derived from the right triangle with legs p and 2c, whose hypotenuse must be of length p + 2a from the focal definition.

As the other conic sections, the hyperbola has conjugate diameters. To exhibit them, we need the conjugate hyperbola, which is constructed on the same reference rectangle. Its equation is obtained by changing +1 to -1 in the canonical equation, or by interchanging a and b. Its foci F''' and F"" are the same distance c from the origin, so all four foci lie on a circle. Of course, the asymptotes are the same. A diameter, such as AB, is any line passing through O that intersects the two branches of the hyperbola. The conjugate diameter CD is drawn between the points of tangency of lines parallel to the diameter that touch the conjugate hyperbola. The conjugate diameter bisects all these chords (it does not seem so in the sketch, because the curves are not accurate). This property may be used to construct normals and tangents as an alternative to the focal property.

The polar equation of the hyperbola is r = p / (1 + e cos ω), which gives both branches as ω goes from 0 to 2π, one branch corresponding to negative values of r. The asymptotic directions are given by ω = cos-1 (1/e). A parametric equation is x = a cosh t, y = b sinh t, using hyperbolic functions, and another is x = a sec t, y = b tan t. Finally, a hyperbola is the intersection of a cone (really, a double cone extending in both directions) with a plane with an inclination greater than the cone angle.

The draftsman is not often required to draw hyperbolas. It is easiest to draw one from the focal definition. An arc is drawn from F with any radius r, and this is intersected by an arc drawn from F' with radius r - 2a. It is easy to locate the foci when a and b are given, so this process is convenient. The intersections are good near the origin, but become poor farther out on the asymptotes. However, they are not needed here.

The Equilateral Hyperbola

When b = a, a special curve is obtained that bears the same relation to the hyperbola as the circle bears to the ellipse. The reference rectangle becomes a square, and the asymptotes make angles of 45° with the axes, and are perpendicular to each other. This is called the equilateral hyperbola, and all these curves are the same shape, differing only in size. The canonical equation becomes x2 - y2 = a2. If the asymptotes are taken as the coordinate axes, the result is xy = a2/2, or xy = constant, a pleasantly elegant result. Isotherms of an ideal gas, pV = nRT, are equilateral hyperbolas. Other examples of this relationship can be found. Unlike circles, equilateral hyperbolas are not good wheels, and are not as easy to draw.

If one point P is known on an equilateral hyperbola, another P' can be found by the construction sketched at the right. Horizontal and vertical lines AC and PE are drawn through P. Then any point B on AO is chosen, and a horizontal line drawn through B intersecting PE at D. Now a line OC is drawn from the orgin through D to a point C on the horizontal line through P. The intersection of the horizontal through B and a vertical line through C determines the second point P'. We see that AC/OE = AO/BO from similar triangles, so AC · BO = OE · AO, which is just xy = x'y'

An example of an equilateral hyperbola occurring in nature is shown at the left. Two parallel glass plates in contact at the left, and separated by about 5 mm at the right, are dipped in beet juice, which rises by capillarity to form an equilateral hyperbola. This can be shown as follows: if the separation of the plates is d = ax cm, and the surface tension is T dyne/cm, then by equating the upward capillary force to the weight of the fluid supported in a small distance dx, 2Tdx = ρgyaxdx, since the angle of contact is zero. Therefore, xy = 2T/ρga = constant, so the curve is a rectangular hyperbola.


Hyperbolic Functions

The hyperbolic functions mentioned above are combinations of exponentials and their connection with the hyperbola is not obvious. From their names, they are analogous to the trigonometric functions. In fact, hyperbolic functions are related to the unit rectangular hyperbola x2 - y2 = 1 just as the trigonometric functions are related to the unit circle x2 + y2 = 1. If we introduce a parameter t, then the unit circle is expressed by x = cos t, y = sin t. Similarly, the unit hyperbola can be expressed as x = cosh t, y = sinh t. The parameter t in both cases can be interpreted as twice the area swept out by a radius vector from the origin O to a point P on the circle or hyperbola. For the circle, this relation is obvious, since A = (t/2π)(π) = t/2, where π is the area of the unit circle. These relations are shown in the figure. Relations between the functions are easily derived by using the properties of right triangles and the equations of the circle or the hyperbola.

For the hyperbola, we may make the linear substitution x - y = η√2 and x + y = ξ√ that rotates the hyperbola to the first quadrant in the (ξ,η)-plane, where its equation is ξη = 1/2. It is a little tricky to find an easy way to find the area A and show that it equals 2t. The area we are seeking A = area OAPQ - ΔOPQ, while area ABQP is area OAPQ - ΔOAB. The two triangles are of the same area, however, since their areas are ξη/2, which is a constant on the hyperbola. Twice the area is then easily seen to be 2A = 2∫((x+y)/√2,1/√2) dη/2η = ln(x+y) = ln[x ± √(x2 - 1)]. But, cosh t = x so that t = cosh-1x = ln[x ± √(x2 - 1], and so 2A = t, just as for the circular funtions.

The logarithmic function for the inverse of x = cosh t may be surprising. However, since cosh t = (et + e-t)/2 = x, or 2x = u + 1/u with the substitution u = et. The quadratic equation for u has the roots u = x ± √(x2 - 1), from which t = cosh-1 x = ln[x ± √(x2 - 1)]. Similar formulas for sinh-1x and tanh-1x exist. For example, we can write tanh t = x = (u - 1/u)/(u + 1/u), so u2 = (1 + x)/(1 - x), and t = (1/2)ln[(1 + x)/(1 - x)], valid for |x| < 1.

The relation cosh2t - sinh2t = 1 is also easily derived by expressing the hyperbolic functions in terms of exponentials. Every trigonometric relation has a hyperbolic analogue, perhaps differing by a minus sign. To find cosh(a + b), for example, use ea = cosh a + sinh a and eb = cosh b + sinh b in 2cosh(a + b) = eaeb + e-ae-b. Multiply out and combine terms. Of course cosh -a = cosh a and sinh -a = -sinh a. The result is cosh(a + b) = cosh a cosh b + sinh a sinh b. For the derivatives, we find d(cosh x)/dx = sinh x and d(sinh x)/dx = cosh x. By the inverse function rule, d(sinh-1x)/dx = 1/cosh x = 1/√(x2 - 1).

Elliptic functions are not defined analogously to circular and hyperbolic functions, but in terms of certain elliptic integrals, so-called because they solve problems associated with the elllipse, such as arc length.

Hyperbolas and Orbits

Newtonian mechanics tells us that any of the conic sections can be an orbit, and we have investigated the cases of planets (ellipses) and comets (parabolas) in the pages on those curves. The hyperbolic orbit is the path of a particle under an inverse-square force that approaches the center of attraction or repulsion at a finite speed along an asymptote, is deflected, and recedes in the same way along the other asymptote. The effect is to change the direction of motion of the particle, without changing its speed. As always, we consider the center of force to be fixed. All two-body problems can be reduced to this case. Our conclusions apply only to the reference system in which the center is fixed. The motion takes place with constant areal velocity A = h/2, where h is a constant related to the angular momentum.

There are no examples of celestial bodies with hyperbolic orbits about the sun. They are not impossible, merely very unlikely, and probably have occurred from time to time. Anything other than volatile cometary debris would probably not be noticed unless it was quite large and dangerous. Such encounters have been blamed for the Moon, but this is just wild speculation. Hyperbolic orbits could be created within the solar system, by certain types of gravitational encounters, or by rockets, but escaping from the Sun is rather difficult.

Alpha particles are the nuclei of helium atoms, with mass 4 and a positive charge of 2e. They are emitted from certain heavy nuclei, such as Polonium, as they strive to a more stable state, with energies in the MeV (mega-electron-volt) range. They knock electrons out of any atoms near their paths, creating densely ionized paths that can be observed in cloud chambers where they trigger condensation. They exhaust their energy in a few centimeters in air, and in a very short distance in solid materials. They cause ZnS crystals to give a flash of luminescence if they hit them, so they can be observed and counted in a spinthariscope. Modern instrumentation makes their observation and counting much more convenient, but this is all that was available in the early 1900's.

Ernest Rutherford (1871-1937) and his students noted in 1911 that alpha particles passing through very thin gold foils were occasionally scattered through large angles. This is an extraordinary effect, like firing a rifle through a wheat field and having the bullet come back at you. What would be expected were numerous slight deflections by the positive charges distributed through the matter. Electrons were known to be light, and could not produce large deflections, just slight wiggles in the paths (which are observed). To cause large deflections, the positive charge and the mass must be concentrated in very small volumes. Rutherford showed that although atoms have a radius of the order of 10-8 cm, the mass and positive charge are concentrated within a radius of about 10-13 cm. If an atom were the size of the earth, then its nucleus would be a few meters in diameter.

An alpha particle, with charge +2e and mass of 4 amu, would then approach a gold nucleus of charge +79e and mass 197 amu at high velocity, and at an impact parameter of b. Only for small b would there be a considerable deflection, and in this case the electrons could be considered distant and diffuse, with the full nuclear charge effective. The force between charges z and Z is zZe2/4πεr2 in MKSA units, so the trajectory will be a hyperbola. Let K = zZe2/4πεm, where m is the (reduced) mass of the alpha particle. This corresponds to GM = k2 for the gravitational problem. The impact parameter b is just the minor axis of the hyperbola (verify this by drawing a right triangle), while a is determined by the total energy. In fact, a = K/vo2, where vo is the initial velocity. We note that the two branches of a hyperbola correspond to attractive and repulsive orbits.

Knowing a and b, we know the parabola, and can find the angle between the asymptotes, and thus the deflection D. D = π - 2θ, so tan θ = cot (D/2) = bvo2/K. This is the relation between the impact parameter b and the deflection D. The distance of closest approach is q = a (e + 1), where e can be found from a and b. This is how Rutherford determined an upper limit on the size of the nucleus, from the maximum observed deflection of the alpha particles. The solid angle between the axis and the deflection D is Ω = 2π(1 - cos D), while the area within the impact parameter b is A = πb2. Therefore, the area dA for scattering into a solid angle dΩ is given by dA/dΩ = C / sin4(D/2), where C = (zZe2/8πεmvo2)2. This is the famous Rutherford scattering differential cross section, proportional to the inverse fourth power of half the angle of deflection. C is the cross section for scattering directly backwards (D = 180°).

Rutherford's discovery of the nucleus led soon after to Bohr's atom, and from there to quantum mechanics, revealing our modern view of matter. Rutherford received a peerage and a Nobel Prize, which were richly deserved. Discoveries in mathematics and physics may lead to understanding; discoveries in most other sciences lead only to knowledge.

Hyperbolas and the Dew Bow

Light that enters a sphere of water and is reflected from the far side to re-emerge in the direction from which it entered is responsible for the lovely phenomenon of the rainbow. The colors arise near a caustic surface created by the fact that there is an angle of minimum deviation. Near this limit there are two beams that interfere to produce a maximum of intensity. Since the angle differs slightly due to the variation of the index of refraction with wavelength, bright colors are seen. Red is on the outside of the rainbow, and blue is on the inside. There may be a secondary rainbow outside the primary one, with a reversed order of colors, which corresponds to two internal reflections. The space between the primary and secondary rainbows is darker than the space outside, since there is no scattered light in this area. Beyond the blue edge of the rainbow supernumerary arcs are often seen, which alternate green and pink. This is just a brief review of the properties of a variable phenomenon. For more information, see a reliable source such as R. A. R. Tricker, Introduction to Meterorological Optics (New York: American Elsevier, 1970).

The primary rainbow angle is about 42°, as shown in the diagram on the right. Any droplet in the cone of angle 42° with vertex at the eye and axis in the solar direction will send color to the eye, whatever its distance may be. This holds for raindrops a and b, as well as for dewdrop c. The rainbow is familiar and is often seen, especially on summer afternoons, but the dewbow is less often noticed. It appears when looking westward over a lawn on a misty morning. The dewdrops give a brilliant reflection in the direction of the antisolar point, where your head casts a shadow, so you can recognize the axis of the cone clearly. This is the heiligenschein, a different, cat's eye, effect that is not related to the dewbow. There may even be a colored glory if there is a mist. The dewbow is seen between the antisolar point and your station, stretching right and left in a curve along the ground. It is the section of the rainbow cone by the earth, and is, therefore, a hyperbola. The secondary rainbow and supernumerary arcs have not been reported in the dewbow, but they certainly exist under the proper conditions, and are something to look for specially. Dew occurs when the surface has cooled by radiation below the temperature of the air, and below the dew point at which the air is saturated by water.

Rainbow phenomena can also be seen in the droplets produced by lawn sprinklers and hose nozzles, or any other source of water droplets. Surface tension produces accurately spherical droplets, especially with small droplets, where gravitational and aerodynamic forces are negligible.

Miscellaneous Hyperbolas

Hyperbolas are not used in surveying for transition curves between two tangents, that might be considered as asymptotes, because they have no definite start or end, and are difficult to compute and lay out. They are also not used for arches or bridges, since they are not as pleasing to the eye as ellipses, circles and parabolas and offer no structural advantages. They do not occur naturally in terrestrial motion or physical processes. For these reasons, hyperbolas are seldom encountered, except as discussed above.

Suppose there are two radio broadcasting stations that emit waves containing accurate timing information. A ship may receive these signals, and note the time displacement between them, which corresponds to a certain distance depending on the propagation speed of the signals. This defines a hyperbola that can be drawn on the map, since the foci are known. If there is a third station, additional hyperbolas can be drawn and the location of the ship determined by the intersection of the curves, with a valuable check since there is more information than the minimum required. LORAN is an example of such a system. Of course, the time of travel of the signal from the broadcasting station can give circles of position that are easier to draw, but the hyperbola method must be used if the times of emission are not known and only differences can be measured.

In the 17th century, there was great interest in improving telescopes, which were severely limited by the spherical aberration of the spherical refracting surfaces available, which meant that they did not bring parallel rays to a point focus. Descartes worked out the forms of surfaces that would bring rays coming from a point source (or infinity) to a point focus; that is, which would provide stigmatic imaging. These were the famous Cartesian Ovals, and included among them were hyperboloidal surfaces. Their form depended on the index of refraction and on the object and image distances, so they were inflexible in application, but worst, they could not be manufactured with sufficient accuracy. For optical work, the surfaces must be correct to less than a wavelength, and this was simply impossible for other than spherical (or cylindrical) surfaces. It is still largely impossible, though good aspheric optical surfaces can be made in certain cases, such as corrector plates for Schmidt and Maksutov telescopes. Approximate shapes are good enough for non-imaging lenses, such as condensers and illumination, and aspheric surfaces are quite popular for these applications, though the surfaces may or may not be hyperboloids. As it happens, there are other problems besides spherical aberration, such as chromatic aberration, and field of view, that cannot be solved with aspheric surfaces, though several coaxial spherical surfaces (which can be very accurately produced) and glasses of different index and dispersion, can solve the problem very well.


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Composed by J. B. Calvert
Created 8 May 2002
Last revised 1 January 2005