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A parabolic reflector (or dish or mirror) is a reflective device used to collect or project energy such as light, sound, or radio waves. Its shape is that of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis Lenovo S10-2 Battery 5200mAh .

The parabolic reflector transforms an incoming plane wave traveling along the axis into a spherical wave converging toward the focus. Conversely, a spherical wave generated by apoint source placed in the focus is transformed into a plane wave propagating as a collimated beam along the axis  Lenovo Y730 Battery 5200mAh .

Parabolic reflectors are used to collect energy from a distant source (for example sound waves or incoming star light) and bring it to a commonfocal point , thus correcting spherical aberration found in simpler spherical reflectors. Since the principles of reflection are reversible  Lenovo Y710 Battery 5200mAh ,

parabolic reflectors can also be used to project energy of a source at its focus outward in a parallel beam , used in devices such as spotlights and car headlights.

The parabolic reflector functions due to the geometric properties of the paraboloidal shape Lenovo Y530 Battery 5200mAh :

if the angle of incidence to the inner surface of the collector equals the angle of reflection, then any incoming ray that is parallel to the axis of the dish will be reflected to a central point, or "focus". Because many types of energy can be reflected in this way, parabolic reflectors can be used to collect and concentrate energy entering the reflector at a particular angle Lenovo Y510 Battery 5200mAh .

Similarly, energy radiating from the "focus" to the dish can be transmitted outward in a beam that is parallel to the axis of the dish.

In contrast with spherical reflectors, which suffer from a spherical aberration that becomes stronger as the ratio of the beam diameter to the focal distance becomes larger  Lenovo S10-3 Battery 5200mAh ,

the parabolic reflectors can be made to accommodate beams of any width. However, if the incoming beam makes a non-zero angle with the axis (or if the emitting point source is not placed in the focus), parabolic reflectors suffer from an aberration called comaLenovo G450 Battery 5200mAh .

This is primarily of interest in telescopes because most other applications do not require sharp resolution off the axis of the parabola.

History

The principle of parabolic reflectors has been known since classical antiquity, when the mathematician Diocles described them in his book On Burning Mirrorsand proved that they focus a parallel beam to a point  Lenovo E30 Battery 5200mAh .

Archimedes in the third century BC studied paraboloids as part of his study of hydrostatic equilibrium, and it has been claimed that he used reflectors to set the Roman fleet alight during the Siege of Syracuse. This seems unlikely to be true, however, as the claim does not appear in sources before the 2nd century AD, and Diocles does not mention it in his book Lenovo E40 Battery 5200mAh .

Parabolic mirrors were also studied by the physicist Ibn Sahl in the 10th century. James Gregory, in his 1663 book Optica Promota(1663), pointed out that a reflecting telescope with a mirror that was parabolic would correct spherical aberration as well as the chromatic aberration seen in refracting telescopes  Lenovo E50 Battery 5200mAh .

The design he came up with bears his name: the "Gregorian telescope"; but according to his own confession, Gregory had no practical skill and he could find no optician capable of actually constructing one. Isaac Newton knew about the properties of parabolic mirrors but chose a spherical shape for his Newtonian telescope mirror to simplify construction Lenovo G460 Battery 5200mAh .

Lighthouses also commonly used parabolic mirrors to collimate a point of light from a lantern into a beam, before being replaced by more efficient fresnel lenses in the 19th century  Lenovo Y460 Battery 5200mAh .

Applications

The most common modern applications of the parabolic reflector are in satellite dishes, reflecting telescopes, radio telescopes, parabolic microphones, and many lighting devices such as spotlights,car headlights, PAR Cans and LED housings Lenovo Y450 Battery 5200mAh .

The Olympic Flame has been lit using a parabolic reflector concentrating sunlight. Parabolic mirrors are one of many shapes for a burning-glass.

Parabolic mirror wok with optical illusion  Lenovo N200 Battery 5200mAh

A parabolic reflector pointing upward can be formed by rotating a reflective liquid, like mercury, around a vertical axis. This makes the liquid mirror telescopepossible.

A mirascope, mirage bowl  Lenovo T410 Battery 5200mAh ,

or parabolic mirror wok is a toy consisting of two parabolic mirrors that are put together like a wok pan and a lid with a hole in it. The device produces an upright real image (with the magnification 1) of an object placed on the bottom of it, which gives the optical illusion that something is in the hole of the lid (while actually it is at the bottom of the pan below) Lenovo T510 Battery 5200mAh .

Parabolic reflectors are a popular alternative for increasing wireless signal strength. Even with simple ones, users have reported 3 dB or more gains Lenovo SL410 Battery 5200mAh .

In mathematics, the parabola s a conic section, the intersection of a right circularconical surface and a plane parallel to a generating straight line of that surface. Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola  Lenovo SL510 Battery 5200mAh .

The line perpendicular to the directrix and passing through the focus (that is the line that splits the parabola through the middle)is called the "axis of symmetry". The point on the axis of symmetry that intersects the parabola is called the "vertex". The vertex is the point where the curvature is greatest  Lenovo C100 Battery 5200mAh .

Parabolas can open up, down, left, right, or in some other arbitrary direction.

The name "parabola" is derived from a New Latin term that means something similar to "compare" or "balance".

The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles Lenovo T400S Battery 5200mAh .

They are frequently used in physics,engineering, and many other areas.

The earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolae  Lenovo Y430 Battery 5200mAh .

(The solution, however, does not meet the requirements imposed by compass and straightedge construction.) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola  Ibm ThinkPad G40 Battery 5200mAh .

The name "parabola" is due to Apollonius, who discovered many properties of conic sections. The focus–directrix property of the parabola and other conics is due to Pappus.

Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity Ibm ThinkPad X200T Battery 5200mAh .

The idea that a parabolic reflector could produce an image was already well-known before the invention of thereflecting telescope. Designs were proposed in the early to mid seventeenth century by manymathematicians including René Descartes, Marin Mersenne, and James Gregory  Apple A1280 battery 5200mAh .

When Isaac Newtonbuilt the first reflecting telescope in 1668 he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers  Apple A1281 battery 5200mAh .

A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolae are similar, meaning that while they can be different sizes, they are all the same shape. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction Samsung R522 battery 5200mAh .

In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid.

A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix Samsung P467 battery 5200mAh .

The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution.

The parabola is found in numerous situations in the physical world (see below) Hp Pavilion dm3 Battery 5400mAh .

In nature, approximations of parabolae and paraboloids (such as catenary curves) are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance Hp Pavilion dm4 Battery 5200mAh

(for instance, a baseball flying through the air, neglecting air friction).

The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes  Hp Pavilion cq62 Battery 5200mAh .

He also later proved this mathematically in his book Dialogue Concerning Two New Sciences. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola  Hp Pavilion cq42 Battery 5200mAh .

As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola  Hp Pavilion cq72 Battery 5200mAh .

Another situation in which parabolae may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a hyperbola or an ellipse are much more common  Hp cq20 Battery 2600mAh .

In fact, the parabolic orbit is the borderline case between those two types of orbit. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.

Hercilio Luz Bridge, Florianópolis, Brazil  Hp 2230 Battery 2600mAh .

Suspension bridges follow a curve which is intermediate between a parabola and acatenary.

Approximations of parabolae are also found in the shape of the main cables on a simple suspension bridge. The curve of the chains of a suspension bridge is always an intermediate curve between an parabola and a catenary, but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used  Dell Inspiron Mini 12 battery 7800mAh .

Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise hyperbolic cable is deformed toward a parabola. Unlike an inelastic chain, a freely-hanging spring of zero unstressed length takes the shape of a parabola.

Paraboloids arise in several physical situations as well Dell XPS M1730 battery 7800mAh .

The best-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes  Dell XPS M2010 battery 7800mAh ,

who, according to a legend of debatable veracity,  constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th centuryDell Studio XPS 1340 battery 5200mAh/7800mAh .

Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite dish antennas.

Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities are contained in a narrow rectangular tank Dell Inspiron 9100 battery 7800mAh .

Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope Dell Inspiron N4010 battery 5200mAh .

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's “Vomit Comet,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes Dell Inspiron N3610 battery 5200mAh .

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates (x,x^2,x^3,\dots,x^n);the standard parabola is the case n = 2, and the case n = 3 is known as the twisted cubic. A further generalization is given by the Veronese variety, when there are more than one input variable Dell Inspiron N5010 battery 5200mAh .

In the theory of quadratic forms, the parabola is the graph of the quadratic form x2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x2 + y2 (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form x2 ? y2. Generalizations to more variables yield further such objects  Dell Inspiron N7010 battery 5200mAh .

The curves y = xp for other values of p are traditionally referred to as the higher parabolas, and were originally treated implicitly, in the form xp = kyq for p and q both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula y = xp / q for a positive fractional power of x  Dell Inspiron 1464 battery 5200mAh .

Negative fractional powers correspond to the implicit equation xpyq = k, and are traditionally referred to as higher hyperbolas.Analytically, x can also be raised to an irrational power (for positive values of x); the analytic properties are analogous to when x is raised to rational powers  Dell Inspiron 1564 battery 5200mAh ,

but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.

A parabolic antenna is an antenna that uses a parabolic reflector, a surface with the cross-sectional shape of a parabola, to direct the radio waves  Dell Inspiron 1764 battery 5200mAh .

The most common form is shaped like a dish and is popularly called a dish antenna or parabolic dish. The main advantage of a parabolic antenna is that it is highly directive; it functions analogously to a searchlight or flashlight reflector to direct the radio waves in a narrow beam, or receive radio waves from one particular direction only  Dell Inspiron 1320 battery 5200mAh .

Parabolic antennas have some of the highest gains, that is they can produce the narrowest beam width angles, of any antenna type. They are used as high-gain antennas for point-to-point radio, television and data communications, and also for radiolocation (radar), on theUHF and microwave (SHF) parts of the radio spectrum  Dell Inspiron 1320n battery 5200mAh .

The relatively short wavelength of electromagnetic radiation at these frequencies allows reasonably sized reflectors to exhibit the desired highly directional response  Dell Vostro 1710 battery 5200mAh .

With the advent of TVRO and DBS satellite television dishes, parabolic antennas have become a ubiquitous feature of the modern landscape. They are also widely used for terrestrial microwave relay links, ground based and airborne radar antennas, wireless WAN/LAN links, satellite and spacecraft communication antennas, and radio telescopes Dell Vostro 1720 battery 5200mAh .

The idea of using parabolic reflectors for radio antennas was taken from optics, where the power of a parabolic mirror to focus light into a beam has been known since classical antiquity. The designs of some specific types of parabolic antenna, such as the Cassegrain and Gregorian   Dell Studio XPS 1640 battery 5200mAh ,

come from similarly-named analogous types of reflecting telescope, which were invented by astronomers during the 15th century.

German physicist Heinrich Hertz constructed the world's first parabolic reflector antenna in 1888  Dell Vostro A840 battery 5200mAh .

The antenna was a cylindrical parabolic reflector made of zinc sheet metal supported by a wooden frame, and had a spark-gap excited dipole along the focal line. Its aperture was 1.2 meters wide, with a focal length of 0.12 meters, and was used at an operating frequency of about 450 MHz  Dell Vostro A860 battery 5200mAh .

With two such antennas, one used for transmitting and the other for receiving, Hertz demonstrated the existence radio waves which had been predicted by James Clerk Maxwell some 22 years earlier  Dell Studio 1535 battery 5200mAh/7800mAh .

The first parabolic antenna used for satellite communications was established in 1962 at Goonhilly in Cornwall, England, UK and was intended to communicate with Telstar.

Types

Parabolic antennas are distinguished by their shapes  Dell Studio 1735 battery 5200mAh/7800mAh :

  • Paraboloidal or dish - The reflector is shaped like a paraboloid. This is the most common type. It radiates a narrow pencil-shaped beam along the axis of the dish.
    • Shrouded dish - Sometimes a cylindrical metal shield is attached to the rim of the dish  Dell Vostro 1310 battery 5200mAh .
    • The shroud shields the antenna from radiation from angles outside the main beam axis, reducing the sidelobes. It is sometimes used to prevent interference in terrestrial microwave links, where several antennas using the same frequency are located close together  Dell Latitude E6400 battery,10400mAh/7800mAh .
    • The shroud is coated inside with microwave absorbent material. Shrouds can reduce back lobe radiation by 10 dB.
  • Cylindrical - The reflector is curved in only one direction and flat in the other. The radio waves come to a focus not at a point but along a line. The feed is often a dipole antenna located along the focal line Dell Latitude E6400 Battery,10400mAh/7800mAh .
  • It radiates a fan-shaped beam, narrow in the curved dimension, and wide in the uncurved dimension. The curved ends of the reflector are sometimes capped by flat plates, to prevent radiation out the ends, and this is called a pillbox antenna Dell Latitude E6400 Battery,10400mAh/7800mAh .
    • "Orange peel" - Another type is very long and narrow, shaped like the letter "C". This is called an orange peel design, and radiates an even wider fan beam. It is often used for radar antennas Dell Latitude E6500 battery,10400mAh/7800mAh.

They are also classified by the type of feed; how the radio waves are supplied to the antenna:

  • Axial or front feed - This is the most common type of feed, with the feed antenna located in front of the dish at the focus, on the beam axis Dell Latitude E6500 battery,10400mAh/7800mAh .
  • A disadvantage of this type is that the feed and its supports block some of the beam, which limits the aperture efficiency to only 55 - 60%.
  • Offset or off-axis feed - The reflector is an asymmetrical segment of a paraboloid, so the focus, and the feed antenna, are located to one side of the dish  Dell Latitude E6500 battery,10400mAh/7800mAh .
  • The purpose of this design is to move the feed structure out of the beam path, so it doesn't block the beam. It is widely used in home satellite televisiondishes, which are small enough that the feed structure would otherwise block a significant percentage of the signal  Dell Latitude E6500 battery,10400mAh/7800mAh .
  • Cassegrain - In a Cassegrain antenna the feed is located on or behind the dish, and radiates forward, illuminating a convex hyperboloidal secondary reflector at the focus of the dish. The radio waves from the feed reflect back off the secondary reflector to the dish, which forms the main beam  HP Pavilion DV6-1210SA battery - 5200mAh/8800mAh .
  • An advantage of this configuration is that the feed, with its waveguides and "front end" electronics does not have to be suspended in front of the dish, so it is used for antennas with complicated or bulky feeds, such as large satellite communication antennas and radio telescopes  HP Pavilion DV7 battery - 7200mAh .
  • Aperture efficiency is on the order of 65 - 70%
  • Gregorian - Similar to the Cassegrain design except that the secondary reflector is concave, (ellipsoidal) in shape. Aperture efficiency over 70% can be achieved  Ibm THINKPAD X200 Battery 8800mAh .

Design

The operating principle of a parabolic antenna is that a point source of radio waves at the focal point in front of a parabolic reflector will be reflected into a collimated plane wave beam along the axis of the reflector. Conversely, an incoming plane wave parallel to the axis will be focused to a point at the focal point  Ibm THINKPAD X200 Battery 8800mAh .

A typical parabolic antenna consists of a parabolic reflector with a small feed antenna at its focus, pointed back toward the reflector. The reflector is a metallic surface formed into a paraboloid of revolution and usually truncated in a circular rim that forms the diameter of the antenna Ibm THINKPAD X200 Battery 8800mAh .

The reflector dish can be of sheet metal, metal screen, or wire grill construction, and it can be either circular or various other shapes to create different beam shapes. A mesh screen reflects radio waves as well as a solid metal surface as long as the holes are smaller than 1/10 of a wavelength, so screen reflectors are often used to reduce weight and wind loads on the dish SONY VAIO VGN-FW21E Battery .

To achieve the maximum gain, it is necessary that the shape of the dish be accurate within a small fraction of a wavelength, to ensure the waves from different parts of the antenna arrive in phase. Large dishes often require a truss structure behind them to provide the required stiffness  Asus M52N Battery .

The feed antenna at the reflector's focus is typically a low-gain type such as a half-wave dipole or more ofteh a small horn antenna called a feed horn. In more complex designs, such as the Cassegrainand Gregorian, a secondary reflector is used to direct the energy into the parabolic reflector from a feed antenna located away from the primary focal point  Dell Latitude E6400 ATG Battery .

The feed antenna is connected to the associated radio-frequency (RF) transmitting or receiving equipment by means of a coaxial cable transmission line or waveguide.

Feed pattern

The radiation pattern of the feed antenna has a strong influence on the aperture efficiency, which determines the antenna gain (see next section)  SONY VAIO VGN-FW21J Battery .

Radiation from the feed that falls outside the edge of the dish is called "spillover" and is wasted, reducing the gain and increasing the backlobes, possibly causing interference or (in receiving antennas) increasing susceptibility to ground noise  SONY VAIO VGN-FW21L Battery .

However, maximum gain is only achieved when the dish is uniformly "illuminated" with a constant field strength to its edges. So the ideal radiation pattern of a feed antenna would be a constant field strength throughout the solid angle of the dish, dropping abruptly to zero at the edges  SONY VAIO VGN-FW21M Battery .

However, practical feed antennas have radiation patterns that drop off gradually at the edges, so the feed antenna is a compromise between acceptably low spillover and adequate illumination.

Gain

The directive qualities of an antenna are measured by a dimensionless parameter called its gain  SONY VAIO VGN-FW21Z Battery ,

which is the ratio of the power received by the antenna from a source along its beam axis to the power received by a hypothetical isotropic antenna. The gain of a parabolic antenna is:

G = \frac{4 \pi A}{\lambda^2}e_A = \frac{\pi^2d^2}{\lambda^2}e_A

where:  SONY VAIO VGN-FW25T/B Battery

  • A is the area of the antenna aperture, that is, the mouth of the parabolic reflector
  • d is the diameter of the parabolic reflector
  • ? is the wavelength of the radio waves. SONY VAIO VGN-FW26T/B Battery
  • eA is a dimensionless parameter called the aperture efficiency. The aperture efficiency of typical parabolic antennas is 0.55 to 0.70.

It can be seen that, as with any aperture antenna, the larger the aperture is, compared to the wavelength, the higher the gain  SONY VAIO VGN-FW27/B Battery .

The gain increases with the square of the ratio of aperture width to wavelength, so large parabolic antennas, such as those used for spacecraft communication and radio telescopes, can have extremely high gain. Applying the above formula to the 25-meter-diameter antennas used by the VLA and VLBA radio telescopes at a wavelength of 21 cm SONY VAIO VGN-FW27/W Battery

(1.42 GHz, a common radio astronomy frequency) yields an approximate maximum gain of 140,000 times or about 50 dBi (decibels above the isotropiclevel).

As of 2009, the largest "dish" antenna in the world is the Arecibo Observatory's radio telescope  SONY VAIO VGN-FW31ZJ Battery ,

at Arecibo, Puerto Rico, which has a diameter of 1,000 ft. (305 m). It has a gain of about 10 million, or 70 dBi, at 2.38 GHz. The dish is built into a valley in the landscape, so it is not steerable. To steer the beam to different points in the sky, the feed antenna is movedSONY VAIO VGN-FW27T/H Battery .

For this reason, the dish actually has a spherical rather than a parabolic shape. A spherical reflector does not have a single focal "point", however, the Arecibo antenna is a three-reflector variety of Gregorian telescope, and uses its secondary and tertiary reflectors to focus the radio waves to a single point SONY VAIO VGN-FW29/B Battery .

Aperture efficiency eA is a catchall variable which accounts for various losses that reduce the gain of the antenna from the maximum that could be achieved with the given aperture. The major factors reducing the aperture efficiency in parabolic antennas are  SONY VAIO VGN-FW290JRB Battery :.

  • Feed spillover - Some of the radiation from the feed antenna falls outside the edge of the dish and so doesn't contribute to the main beam  SONY VAIO VGN-FW290JTB Battery .
  • Feed illumination taper - The maximum gain for any aperture antenna is only achieved when the intensity of the radiated beam is constant across the aperture area  SONY VAIO VGN-FW290JTH Battery .
  • However the radiation pattern from the feed antenna usually tapers off toward the outer part of the dish, so the outer parts of the dish are "illuminated" with a lower intensity of radiation  SONY VAIO VGN-FW33G/E1 Battery .
  • Even if the feed provided constant illumination across the angle subtended by the dish, the outer parts of the dish are farther away from the feed antenna than the inner parts, so the intensity would drop off with distance from the center  SONY VAIO VGN-FW33G/W Battery .
  • So the intensity of the beam radiated by a parabolic antenna is maximum at the center of the dish and falls off with distance from the axis, reducing the efficiency  SONY VAIO VGN-FW33GW Battery .
  • Aperture blockage - In axis-fed parabolic dishes where the feed antenna is located in front of the dish in the beam path, the feed structure and its supports block some of the beam. In small dishes such as home satellite dishes, where the size of the feed structure is comparable with the size of the dish, this can seriously reduce the antenna gain (Dell XPS M1210 Batteryhttp://www.hdd-shop.co.uk .
  • To prevent this problem these types of antennas often use an off-axis feed, where the feed antenna is located to one side, outside the beam area. The aperture efficiency for these types of antennas can reach 0.7 to 0.8 Dell KM958 battery .
  • Shape errors - random surface errors in the shape of the reflector reduce efficiency. The loss is approximated by Ruze's Equation  Dell Studio 1555 battery .

For theoretical considerations of mutual interference (at frequencies between 2 and c. 30 GHz - typically in the Fixed Satellite Service) where specific antenna performance has not been defined, areference antenna based on Recommendation ITU-R S.465 is used to calculate the interference, which will include the likely sidelobes for off-axis effects Sony VGP-BPS13 battery .

A parabolic trough is a type of solar thermal energy collector. It is constructed as a long parabolic mirror (usually coated silver or polished aluminum) with a Dewar tube running its length at the focal point. Sunlight is reflected by the mirror and concentrated on the Dewar tube  Sony VGP-BPS13/B battery .

The trough is usually aligned on a north-south axis, and rotated to track the sun as it moves across the sky each day.

Alternatively the trough can be aligned on an east-west axis, this reduces the overall efficiency of the collector, due to cosine loss, but only requires the trough to be aligned with the change in seasons, avoiding the need for tracking motors Sony VGP-BPS13/S battery .

This tracking method works correctly at the spring and fallequinoxes with errors in the focusing of the light at other times during the year (the magnitude of this error varies throughout the day, taking a minimum value at solar noon). There is also an error introduced due to the daily motion of the sun across the sky, this error also reaches a minimum at solar noon Sony VGP-BPS13A/B battery .

Due to these sources of error, seasonally adjusted parabolic troughs are generally designed with a lower solar concentration ratio. In order to increase the level of alignment, some measuring devices have also been invented.

Heat transfer fluid (usually oil) runs through the tube to absorb the concentrated sunlight Sony VGP-BPS13B/B battery .

This increases the temperature of the fluid to some 400°C. The heat transfer fluid is then used to heat steam in a standard turbine generator. The process is economical and, for heating the pipe, thermal efficiency ranges from 60-80%. The overall efficiency from collector to grid, i.e  Sony VGP-BPL9 battery .

(Electrical Output Power)/(Total Impinging Solar Power) is about 15%, similar to PV (Photovoltaic Cells) but less than Stirling dish concentrators.

Current commercial plants utilizing parabolic troughs are hybrids; fossil fuels are used during night hours  Sony VGP-BPS13B/B battery ,

but the amount of fossil fuel used is limited to a maximum 27% of electricity production, allowing the plant to qualify as a renewable energy source. Because they are hybrids and include cooling stations, condensers, accumulators and other things besides the actual solar collectors, the power generated per square meter of area varies enormously Sony VGP-BPL15 battery .

Types of mirrors

Usually, mirrors are used which are parabolic and are of a single piece. In addition, V-type parabolic troughs exist which are made from 2 mirrors and placed at an angle towards each other  Dell Inspiron E1505 battery .

Mirror coatings

In 2009, scientists at the National Renewable Energy Laboratory (NREL) and SkyFuel teamed to develop large curved sheets of metal that have the potential to be 30% less expensive than today's best collectors of concentrated solar power by replacing glass-basedHP Pavilion dv6000 Battery .

models with a silver polymer sheet that has the same performance as the heavy glass mirrors, but at a much lower cost and much lower weight. It also is much easier to deploy and install. The glossy film uses several layers of polymers, with an inner layer of pure silver HP Pavilion dv8000 battery .

Energy storage

As this renewable source of energy is inconsistent by nature, methods for energy storage have been studied, for instance the single-tank (thermocline) storage technology for large-scale solar thermal power plants HP PAVILION DV6000 Battery,  8800mAh .

The thermocline tank approach uses a mixture of silica sand and quartzite rock to displace a significant portion of the volume in the tank. Then it is filled with the heat transfer fluid, typically a molten nitrate salt HP PAVILION DV8000 Battery,  8800mAh .

Existing plants

The largest operational solar power system at present is one of the SEGS plants and is located at Kramer Junction in California, USA, with five fields of 33 MW generation capacity each  SONY VAIO VGN-FZ Battery .

The 64 MW Nevada Solar One also uses this technology. In the new Spanish plant, Andasol 1 solar power station, the 'Eurotrough'-collector is used. This plant went online in November 2008  and has a nominal output of 49.9 MW  HP Pavilion DV6000 battery - 8800mAh .

A curved mirror is a mirror with a curved reflective surface, which may be either convex (bulging outward) or concave (bulging inward). Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices  HP Pavilion DV8000 battery - 7800mAh .

The most common non-spherical type areparabolic reflectors, found in optical devices such as reflecting telescopes that need to image distant objects, since spherical mirror systems suffer fromspherical aberration  HP Pavilion DV6000 battery - 8800mAh .

A convex mirror, fish eye mirror or diverging mirror, is a curved mirror in which the reflective surface bulges toward the light source. Convex mirrors reflect light outwards, therefore they are not used to focus light  HP Pavilion DV8000 battery - 6600mAh .

Such mirrors always form a virtual image, since the focus (F) and the centre of curvature (2F) are both imaginary points "inside" the mirror, which cannot be reached. Therefore images formed by these mirrors cannot be taken on screen. (As they are inside the mirror)  SONY VAIO VGN-FZ18 Battery

A collimated (parallel) beam of light diverges (spreads out) after reflection from a convex mirror, since the normal to the surface differs with each spot on the mirror  SONY VAIO VGN-FZ21E Battery .

Uses

The passenger-side mirror on a car is typically a convex mirror. In some countries, these are labeled with the safety warning "Objects in mirror are closer than they appear", to warn the driver of the convex mirror's distorting effects on distance perception  SONY VAIO VGN-FW31J Battery - 5200mAh/7800mAh/10400mAh .

Convex mirrors are preferred in vehicles because they give an upright, though diminished, image. Also they provide a wider field of view as they are curved outwards.

Convex mirrors are used in some automated teller machines as a simple and handy security feature, allowing the users to see what is happening behind them  SONY VAIO VGN-FW31J Battery - 5200mAh/7800mAh/10400mAh .

Similar devices are sold to be attached to ordinary computer monitors.

Some camera phones use convex mirrors to allow the user to correctly aim the camera while taking a self-portrait  SONY VAIO VGN-FW21E Battery .

Image

The image is always virtual (rays haven't actually passed through the image,their extensions do), diminished (smaller), and upright . These features make convex mirrors very useful  SONY VAIO VGN-NR11S/S Battery :

everything appears smaller in the mirror, so they cover a wider field of view than a normal plane mirror does as the image is "compressed".

A toroidal mirror is a form of parabolic reflector which has a different focal distance depending on the angle of the mirror SONY VAIO VGN-NR11M/S Battery .

The curvature is actually that of an elliptic paraboloid with a \ne b. If the shape were that of a toroid, the mirror would exhibit spherical aberration.

Toroidal mirrors are used in Yolo telescopes and optical monochromators (mirrors C and E in the diagram)  SONY VAIO VGN-NR11Z/S Battery .

In these devices, the source and detectors of the light are not located on the optic axis of the mirror, so the use of a true paraboloid of revolution would cause a distorted image.

A solar furnace is a structure that captures sunlight to produce high temperatures, usually for industry  SONY VAIO VGN-NR11Z/T Battery .

This is done with a curved mirror (or an array of mirrors) that acts as a parabolic reflector, concentrating light (Insolation) onto a focal point. The temperature at the focal point may reach 3,500 °C (6,330 °F), and this heat can be used to generate electricity, melt steel, make hydrogen fuel or nanomaterials  Sony VAIO VGN-FZ21E Battery .

The term "solar furnace" has also evolved to refer to solar concentrator heating systems using parabolic mirrors or heliostatswhere 538 °C (1,000 °F) is now commonly achieved. The largest solar furnace is at Odeillo in the Pyrenees-Orientales inFrance, opened in 1970  Sony VAIO VGN-FW21E Battery .

It employs an array of plane mirrors to gather sunlight, reflecting it onto a larger curved mirror. The rays are then focused onto an area the size of a cooking pot and can reach 3,500 °C (6,330 °F), depending on the process installed, for example  Sony VAIO VGN-NR11S/S Battery :

  • about 1,000 °C (1,830 °F) for metallic receivers producing hot air for the next generation solar towers as it will be tested at the Themis plant with the Pegase project
  • about 1,400 °C (2,550 °F) to produce hydrogen by cracking methane molecules Sony VAIO VGN-NR11Z/S Battery
  • up to 2,500 °C (4,530 °F) to test materials for extreme environment such as nuclear reactors or space vehicle atmospheric reentry
  • up to 3,500 °C (6,330 °F) to produce nanomaterials by solar induced sublimation and controlled cooling, such as carbon nanotubes[3] or zinc nanoparticles Sony VAIO VGN-NR11M/S Battery .

History

The ancient Greek / Latin term heliocaminus literally means "solar furnace" and refers to a glass-enclosed sunroom intentionally designed to become hotter than the outside air temperature  Sony VAIO VGN-NR11Z/T Battery .

During the Second Punic War (218 - 202 BC), the Greek scientist Archimedes is said to have repelled the attacking Roman ships by setting them on fire with a "burning glass" that may have been an array of mirrors. An experiment to test this theory was carried out by a group at the Massachusetts Institute of Technology in 2005   SONY VAIO VGN-FZ180E Battery .

It concluded that although the theory was sound for stationary objects, the mirrors would not likely have been able to concentrate sufficient solar energy to set a ship on fire under battle conditions.

The first modern solar furnace is believed to have been built in France in 1949 by Professor Félix Trombe SONY VAIO VGN-FZ220E Battery .

It is now still in place at Mont Louis, near Odeillo. The Pyrenees were chosen as the site due to clear skies up to 300 days a year.

Modern uses

The solar furnace principle is being used to make inexpensive solar cookers and solar-powered barbecues, and for solar water pasteurization  SONY VAIO VGN-FZ340E Battery .

A prototype Scheffler reflector is being constructed in India for use in a solar crematorium. This 50 m² reflector will generate temperatures of 700 °C (1,292 °F) and displace 200-300 kg of firewood used per cremation.

It has been suggested that solar furnaces could be used in space to provide energy for manufacturing purposes SONY VAIO VGN-FZ430E Battery .

Their reliance on sunny weather is a limiting factor as a source of renewable energy on Earth but could be tied to thermal energy storage systems for energy production through these periods and into the night SONY VAIO VGN-FZ460E Battery .

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant SONY VAIO VGN-FZ480E Battery .

Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas SONY VAIO VGN-FZ4000 Battery .

Ellipses also arise as images of a circle underparallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency  SONY VAIO VGN-FZ31E Battery .

An ellipse is a smooth closed curve which is symmetric about its horizontal and vertical axes. The distance between antipodalpoints on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis ortransverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter SONY VAIO VGN-FZ31B Battery .

The semimajor axis (denoted by a in the figure) and the semiminor axis (denoted by b in the figure) are one half of the major and minor diameters, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes,the major and minor semiaxes, or major radius and minor radius  SONY VAIO VGN-FZ31J Battery .

The foci of the ellipse are two special points F1 and F2 on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major diameter (PF1 + PF2 = 2a ). Each of these two points is called a focus of the ellipse  SONY VAIO VGN-FZ31M Battery .

Refer to the lower Directrix section of this article for a second equivalent construction of an ellipse.

The eccentricity of an ellipse, usually denoted by ? or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a  SONY VAIO VGN-FZ31Z Battery .

For an ellipse the eccentricity is between 0 and 1 (0<e<1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape  SONY VAIO VGN-FZ38M Battery .

It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away.
The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse (f = ae) SONY VGP-BPS8 Battery .

An ellipse can be drawn using two drawing pins, a length of string, and a pencil:

Push the pins into the paper at two points, which will become the ellipse's foci. Tie the string into a loose loop around the two pins  SONY VGP-BPS13/S Battery .
Pull the loop taut with the pencil's tip, so as to form a triangle. Move the pencil around, while keeping the string taut, and its tip will trace out an ellipse. Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed; thus it is called the gardener's ellipse  SONY VGP-BPS13A/B Battery .

If the ellipse is to be inscribed within a specified rectangle, tangent to its four sides at their midpoints, one must first determine the position of the foci and the length of the string loop  SONY VGP-BPS13B/B Battery :

Let A,B,C,D be the corners of the rectangle, in clockwise order, with A-B being one of the long sides. Draw a circle centered on A, whose radius is the short side A-D. From corner B draw a tangent to the circle. The length L of this tangent is the distance between the foci   SONY VGP-BPS13A/S Battery .
This length L can be calculated with the Pythagorean theorem. As the tangent is at a right angles to the radius at the intersect of the tangent with the circle L equals square root ((A-B)squared - (A-D)squared) i.e. square root of the long side of the rectangle squared minus short side squared  SONY VGP-BPS13AS Battery .
Draw a horizontal line through the center of the rectangle. This will be the major axis of the ellipse. Place the foci on the major axis, at distance L/2 from the center  Dell Inspiron 1320n Battery .

To adjust the length of the string loop, insert a pin at one focus, and the second pin at the opposite side of the rectangle on the major axis. Loop the string around the two pins and tie it taut. Move the second pin to the other focus. Then draw the ellipse as above; it should fit snugly in the original rectangle  Dell Studio 1450 Battery .

Unfortunately strings tend to be elastic so if you push harder on the pencil stretching the string more you will get a bigger ellipse, pushing less it will be smaller. It may take a few tries to push just hard enough to make the ellipse fit the rectangle Dell Studio 1457 Battery .

In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola (though mathematically quite different)  Dell Latitude D610 Battery .

Its surface of revolution,

the catenoid, is a minimal surface and is the shape assumed by a soap film bounded by two parallel circular rings.

The word catenary is derived from the Latin word catena, which means "chain"  Toshiba NB100 Battery .

Huygens first used the term catenaria in a letter to Leibniz in 1690. However, Thomas Jefferson is usually credited with the English word catenary. The curve is also called the "alysoid", "chainette", or, particularly in the material sciences, "funicular"  Toshiba Satellite M65 battery .

It is often stated that Galileo thought that the curve followed by a hanging chain is a parabola. A careful reading of his book Two new sciences shows this to be an oversimplification. Galileo discusses the catenary in two places; in the dialog of the Second Day he states that a hanging chain resembles a parabola  Toshiba Satellite M60 battery .

But later, in the dialog of the Fourth Day, he gives more details, and states that a hanging cord is approximated by a parabola, correctly observing that this approximation improves as the curvature gets smaller and is almost exact when the elevation is less than 45o  Dell Latitude D830 Battery .

That the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657) and published posthumously in 1669.

The application of the catenary to the construction of arches is due to Robert Hooke, who discovered it in the context of the rebuilding of St Paul's Cathedral, possibly having seen Huygens' work on the catenary Dell Latitude D620 Battery .

(Some much older arches are also approximate catenaries.)

In 1671, Hooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin anagram in an appendix to hisDescription of Helioscopes  Dell Inspiron Mini 10 Battery ,

where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution of this anagram in his lifetime, but in 1705 his executor provided it as Ut pendet continuum flexile, sic stabit contiguum rigidum inversum, meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch Sony VGN-FW11S Battery ."

In 1691 Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli. David Gregory wrote a treatise on the catenary in 1697.

Euler proved in 1744 that the catenary is the curve which, when rotated about the x-axis, gives the surface of minimum surface area (the catenoid) for the given bounding circle  Sony VGN-FW11M Battery .

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows  Sony VGN-FW139E/H battery .

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)  Dell Latitude E5400 Battery .

Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a 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 conic is a hyperbola Dell Latitude E4200 Battery .

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 circleviewed 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 and hence elliptical orbit) Dell Inspiron 300M Battery ,

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) Dell Studio 1737 battery ,

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  Dell Inspiron E1505 battery .

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  Dell RM791 battery .

In the case of the curve f(x) = 1 / x the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipse's 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  Toshiba Satellite P10 Battery .

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 non-Euclidean geometry used in both relativity and quantum mechanics) Acer Aspire One battery 10400mAh .

The word "hyperbola" derives from the Greek ????????, 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 (ca. 262 BC–ca. 190 BC) in his definitive work on the conic sections, the Conics  SONY VGN-FZ210CE Battery .

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), respectively  Dell Precision M70 Battery .

Nomenclature

Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipsedoes. A hyperbola consists of two disconnected curves called its arms or branches Toshiba Satellite L305 Battery .

The closest distance of the two branches connecting their 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 sometimes called the semi-major axis Toshiba Satellite T4900 Battery .

Outside of the transverse axis 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 bisection of the transverse axis. The hyperbola has mirror symmetryabout its principal axes, and is also symmetric under a 180° turn about its center  Toshiba PA3399U-2BRS battery .

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  Sony VGP-BPS10 battery  5200mAh/7800mAh/10400mAh . Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis of aCartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±b?a, where b is given by ? = arctan(b?a) and where the ? is the angle between the transverse axis and either asymptote Sony VGP-BPS9 battery  5200mAh/7800mAh/10400mAh .

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 other principal axis

Sony VGP-BPS11 battery  5200mAh/7800mAh/10400mAh

Sony VGP-BPS11 Battery

; 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 Sony VGP-BPS22 battery  5200mAh/7800mAh/10400mAh .

If b = a, the angle 2? 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 Sony VGP-BPL8 battery  5200mAh/7800mAh/10400mAh .

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

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

A hyperbola aligned in this way is sometimes called an "East-West opening hyperbola" Sony VGP-BPS14 battery  5200mAh/7800mAh/10400mAh ;

by analogy, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola".

The shape of a hyperbola is defined entirely by its eccentricity ?, which is a dimensionless number always greater than one. The distance c from the center to the foci equals a? Sony VGP-BPS21 battery  5200mAh/7800mAh/10400mAh .

The eccentricity can also be defined as the ratio of the distances to either focus and to a corresponding line known as thedirectrix; hence, the distance from the center to the directrices equals a/?. In terms of the parametersa, b, c and the angle ?, the eccentricity equals Toshiba Satellite A200 Battery

\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 (? = 45°, a = b) equals the square root of two, ? =   Toshiba Satellite 1200 Battery .

Every hyperbola has a conjugate hyperbola, in which the transverse and conjugate axes are exchanged without changing the asymptotes. This corresponds to exchanging a and b in the formulae describing the hyperbola; for example, the angle ? of the conjugate hyperbola equals 90° minus the angle of the original hyperbola  Toshiba Satellite M300 Battery .

Thus, unless ? = 45° (a rectangular hyperbola), the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities. Hence, the conjugate hyperbola does not correspond to a 90° rotation of the original hyperbola; the two hyperbolas are generally different in shape  SONY VGP-BPS13A/Q Battery .

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} SONY VGP-BPS13A/Q Battery .

The semi-latus rectum lis 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{l}{\varepsilon} SONY VGP-BPS13B/Q Battery .

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 SONY VGP-BPS13B/Q Battery .

If the plane is parallel to the axis of the double cone and passes through its central apex, a degenerate hyperbola results that is simply two straight lines that cross at the apex point SONY VGP-BPS13/Q Battery .

Difference of distances to foci

A hyperbola may be defined equivalently as the locus of points where the difference of the distances to the two foci is a constant equal to 2a, the distance between its two vertices SONY VGP-BPS13/Q Battery .

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 SONY VGP-BPS21A/B Battery .

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 SONY VGP-BPS21/S Battery ;

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 SONY VGP-BPS21B Battery .

, the angle F1P F2. Consequently, the feet of perpendiculars drawn from each focus to such a tangent line lies on a circle of radius athat is centered on the hyperbola's own center SONY VGP-BPS21A Battery .

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 SONY VGP-BPS21 Battery .

Directrix

The locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola  Sony VGP-BPS21A/B Battery .

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 Sony VGP-BPS21/S Battery .

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 Sony VGP-BPS21A Battery .

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 Sony VGP-BPS21 Battery .

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 Sony Vaio PCG-5G2L Battery .

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 Sony Vaio PCG-5G3L Battery .

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 Sony Vaio PCG-5J1L Battery .

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 Sony Vaio PCG-5K2L Battery .

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 Sony Vaio PCG-5J2L Battery .

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) Sony Vaio PCG-5K1L Battery .

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  Sony Vaio PCG-6S2L Battery .

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  Sony Vaio PCG-6S3L Battery ,

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  Sony Vaio PCG-6V1L Battery .

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  Sony Vaio PCG-6W1L Battery .

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 viewerSony Vaio PCG-6W2L Battery .

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 Sony Vaio PCG-6W3L Battery .

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 parabola Sony Vaio PCG-7111L Battery .

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 Sony Vaio PCG-7112L Battery .

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 Sony Vaio PCG-7113L Battery ,

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 viewerSony Vaio PCG-7133L Battery .

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  Sony Vaio PCG-7Z2L Battery .

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  Sony Vaio PCG-8Y1L Battery .

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 Sony Vaio PCG-8Y2L Battery .

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) Sony Vaio PCG-8Z1L Battery .

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 Sony Vaio PCG-8Z2L Battery .

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 SONY VAIO VGN-FW31J Battery .

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 SONY VAIO VGN-FW31E Battery .

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 SONY VAIO VGN-FW31B Battery ,

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.

A family of confocal hyperbolas is the basis of the system of elliptic coordinates in two dimensions SONY VAIO VGN-FW31M Battery .

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 ? is the angle of the asymptotes with the x-axisSONY VAIO VGN-FW31Z Battery .

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 + ivare those after the transformation SONY VAIO VGN-FW38M Battery .

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 systeminto two families of orthogonal hyperbolas SONY VAIO VGN-AW41JF Battery .

Applications

Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light   SONY VAIO VGN-AW41JF/H Battery .

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  SONY VAIO VGN-AW41MF Battery .

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  SONY VAIO VGN-AW41MF/H Battery .

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  SONY VAIO VGN-FW30B Battery .

Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from aLORAN 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  SONY VAIO VGN-FW50B Battery ;

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 SONY VAIO VGN-FW51B/W Battery .

The paths followed by any particle in the classical Kepler problem is a conic section. In particular, if the total energy E of the particle is greater than zero (i.e., if the particle is unbound), the path of such a particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering high-energy particles SONY VAIO VGN-FW51MF Battery ;

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 SONY VAIO VGN-FW51MF/H Battery .

As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, an intensely studied problem of geometry. Given an angle, one first draws a circle centered on its middle point O, which intersects the legs of the angle at points A and B SONY VAIO VGN-FW51ZF Battery .

One next draws the line through A and B and constructs a hyperbola of eccentricity ?=2 with that line as its transverse axis and B as one focus. The directrix of the hyperbola is the bisector of AB, and for any point P on the hyperbola, the angle ABP is twice as large as the angle BAP SONY VAIO VGN-FW51ZF/H Battery .

Let P be a point on the circle. By the inscribed angle theorem, the corresponding center angles are likewise related by a factor of two, AOP = 2×POB. But AOP+POB equals the original angle AOB. Therefore, the angle has been trisected, since 3×POB = AOB SONY VAIO VGN-FW52JB Battery .

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 SONY VAIO VGN-FW54FB Battery;

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, respectivelySONY VAIO VGN-FW70DB Battery .

Extensions

The three-dimensional analog of a hyperbola is a hyperboloid. Hyperboloid come in two varieties, those of one sheet and those of two sheets SONY VAIO VGN-FW71DB/W Battery .

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  SONY VAIO VGN-FW72JGB Battery .

In physics and geometry, the troposkein is the curve that an idealized rope assumes when anchored at its ends and spun around its long axis at a constantangular velocity. This shape is very similar to the shape assumed by a skipping rope and is independent of rotational speed in the absence of gravity SONY VAIO VGN-FW73JGB Battery ,

but varies with respect to rotational speed in the presence of gravity. The troposkein does not have a closed-form representation; however, in the absence of gravity it can be approximated by a pair of line segments spanned by a circular arc (tangential to the line segments at its endpoints)  SONY VAIO VGN-FW74FB Battery .

The form of a troposkein can be approximated for a given gravational acceleration, rope density and angular velocity by iterative approximation. This shape is also useful for decreasing the stress experienced by the blades of a darrieus vertical axis wind turbineSONY VAIO VGN-FW81HS Battery .

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, andinvolutes  SONY VAIO VGN-FW81NS Battery .

Roughly speaking, it is the curve described by a point (called the generator or pole) attached to a given curve as it rolls without slipping along a second given curve that is fixed SONY VAIO VGN-FW81S Battery .

More precisely, given a curve attached to a plane which is moving so that the curve rolls without slipping along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve in the fixed plane called a roulette SONY VAIO VGN-FW82DS Battery .

In the illustration, the fixed curve (blue) is a parabola, the rolling curve (green) is an equal parabola, and the generator is the vertex of the rolling parabola which describes the roulette (red). In this case the roulette is the cissoid of Diocles  SONY VAIO VGN-FW82JS Battery .

In the case where the rolling curve is a line and the generator is a point on the line, the roulette is called an involute of the fixed curve. If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid. If, in this case, the point lies on the circle then the roulette is a cycloid SONY VAIO VGN-FW82XS Battery .

If, instead of a single point being attached to the rolling curve, another given curve is carried along the moving plane, a family of congruent curves is produced. The envelope of this family may also be called a roulette  SONY VAIO VGN-FW83DS Battery .

A related concept is a glissette, the curve described by a point attached to a given curve as it slides along two (or more) given curves.

Formally speaking, the curves must be differentiable curves in the Euclidean plane SONY VAIO VGN-FW83JS Battery .

One is kept invariant; the other is subjected to acontinuous congruence transformation such that at all times the curves are tangent at a point of contact that moves with the same speed when taken along either curve. The resulting roulette is formed by the locus of the generator subjected to the same set of congruence transformations SONY VAIO VGN-FW83XS Battery .

Modelling the original curves as curves in the complex plane, let r,f:\mathbb R\to\mathbb C be differentiable parametrisations such that r(0)=f(0), r?(0)=f?(0), and |r?(t)|=|f?(t)|?0 for all t. The roulette of p\in\mathbb C as ris rolled on f is then given by the mapping SONY VAIO VGN-FW90NS Battery :

Roulettes in higher spaces can certainly be imagined but one needs to align more than just the tangents.

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