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Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects show a larger parallax than farther objects when observed from different positions, so parallax can be used to determine distances. To measure large distances, such as the distance of a planet or a star from
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
, astronomers use the principle of parallax. Here, the term ''parallax'' is the semi-angle of inclination between two sight-lines to the star, as observed when Earth is on opposite sides of the Sun in its orbit. These distances form the lowest rung of what is called "the
cosmic distance ladder The cosmic distance ladder (also known as the extragalactic distance scale) is the succession of methods by which astronomers determine the distances to celestial objects. A ''direct'' distance measurement of an astronomical object is possible o ...
", the first in a succession of methods by which astronomers determine the distances to celestial objects, serving as a basis for other distance measurements in astronomy forming the higher rungs of the ladder. Parallax also affects optical instruments such as rifle scopes,
binoculars Binoculars or field glasses are two refracting telescopes mounted side-by-side and aligned to point in the same direction, allowing the viewer to use both eyes (binocular vision) when viewing distant objects. Most binoculars are sized to be held ...
,
microscope A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisi ...
s, and twin-lens reflex cameras that view objects from slightly different angles. Many animals, along with humans, have two eyes with overlapping visual fields that use parallax to gain depth perception; this process is known as stereopsis. In
computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
the effect is used for computer stereo vision, and there is a device called a parallax rangefinder that uses it to find the range, and in some variations also altitude to a target. A simple everyday example of parallax can be seen in the dashboards of motor vehicles that use a needle-style mechanical speedometer. When viewed from directly in front, the speed may show exactly 60, but when viewed from the passenger seat, the needle may appear to show a slightly different speed due to the angle of viewing combined with the displacement of the needle from the plane of the numerical dial.


Visual perception

As the eyes of humans and other animals are in different positions on the head, they present different views simultaneously. This is the basis of stereopsis, the process by which the brain exploits the parallax due to the different views from the eye to gain depth perception and estimate distances to objects. Animals also use ''motion parallax'', in which the animals (or just the head) move to gain different viewpoints. For example, pigeons (whose eyes do not have overlapping fields of view and thus cannot use stereopsis) bob their heads up and down to see depth. The motion parallax is exploited also in wiggle stereoscopy, computer graphics that provide depth cues through viewpoint-shifting animation rather than through binocular vision.


Astronomy

Parallax arises due to a change in viewpoint occurring due to the motion of the observer, of the observed, or both. What is essential is relative motion. By observing parallax,
measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s, and using
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, one can determine
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
.


Stellar parallax

Stellar parallax created by the relative motion between the Earth and a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
can be seen, in the Copernican model, as arising from the orbit of the Earth around the Sun: the star only ''appears'' to move relative to more distant objects in the sky. In a geostatic model, the movement of the star would have to be taken as ''real'' with the star oscillating across the sky with respect to the background stars. Stellar parallax is most often measured using annual parallax, defined as the difference in position of a star as seen from the Earth and Sun, i.e. the angle subtended at a star by the mean radius of the Earth's orbit around the Sun. The parsec (3.26
light-year A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distance, astronomical distances and is equivalent to about 9.46 Orders of magnitude (numbers)#1012, trillion kilometers (), or 5.88  ...
s) is defined as the distance for which the annual parallax is 1 
arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...
. Annual parallax is normally measured by observing the position of a star at different times of the
year A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the h ...
as the Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars. The first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61  Cygni using a heliometer.. Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from the Earth to the Sun, now based on
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
reflection off the surfaces of planets. The angles involved in these calculations are very small and thus difficult to measure. The nearest star to the Sun (and thus the star with the largest parallax), Proxima Centauri, has a parallax of 0.7687 ± 0.0003  arcsec. This angle is approximate that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. The fact that stellar parallax was so small that it was unobservable at the time was used as the main scientific argument against heliocentrism during the early modern age. It is clear from
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed entirely implausible: it was one of Tycho's principal objections to Copernican heliocentrism that for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn (then the most distant known planet) and the eighth
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
(the fixed stars). In 1989, the satellite Hipparcos was launched primarily for obtaining improved parallaxes and proper motions for over 100,000 nearby stars, increasing the reach of the method tenfold. Even so, Hipparcos was only able to measure parallax angles for stars up to about 1,600
light-year A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distance, astronomical distances and is equivalent to about 9.46 Orders of magnitude (numbers)#1012, trillion kilometers (), or 5.88  ...
s away, a little more than one percent of the diameter of the Milky Way Galaxy. The
European Space Agency , owners = , headquarters = Paris, Île-de-France, France , coordinates = , spaceport = Guiana Space Centre , seal = File:ESA emblem seal.png , seal_size = 130px , image = Views in the Main Control Room (120 ...
's Gaia mission, launched in December 2013, can measure parallax angles to an accuracy of 10
microarcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...
s, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light-years from Earth. In April 2014, NASA astronomers reported that the
Hubble Space Telescope The Hubble Space Telescope (often referred to as HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the first space telescope, but it is one of the largest and most vers ...
, by using spatial scanning, can precisely measure distances up to 10,000 light-years away, a ten-fold improvement over earlier measurements.


Distance measurement

Distance measurement by parallax is a special case of the principle of triangulation, which states that one can solve for all the sides and angles in a network of triangles if, in addition to all the angles in the network, the length of at least one side has been measured. Thus, the careful measurement of the length of one baseline can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side (the motion of the observer) and the small top angle (always less than 1 
arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...
, leaving the other two close to 90  degrees), the length of the long sides (in practice considered to be equal) can be determined. Assuming the angle is small (see derivation below), the distance to an object (measured in parsecs) is the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of the parallax (measured in
arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...
s): d (\mathrm) = 1 / p (\mathrm). For example, the distance to Proxima Centauri is 1/0.7687 = .


Diurnal parallax

''Diurnal parallax'' is a parallax that varies with the rotation of the Earth or with a difference in location on the Earth. The Moon and to a smaller extent the terrestrial planets or
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
s seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars. The diurnal parallax has been used by John Flamsteed in 1672 to measure the distance to
Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...
at its
opposition Opposition may refer to: Arts and media * ''Opposition'' (Altars EP), 2011 EP by Christian metalcore band Altars * The Opposition (band), a London post-punk band * '' The Opposition with Jordan Klepper'', a late-night television series on Com ...
and through that to estimate the
astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
and the size of the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
.


Lunar parallax

''Lunar parallax'' (often short for ''lunar horizontal parallax'' or ''lunar equatorial horizontal parallax''), is a special case of (diurnal) parallax: the Moon, being the nearest celestial body, has by far the largest maximum parallax of any celestial body, at times exceeding 1 degree. The diagram for stellar parallax can illustrate lunar parallax as well if the diagram is taken to be scaled right down and slightly modified. Instead of 'near star', read 'Moon', and instead of taking the circle at the bottom of the diagram to represent the size of the Earth's orbit around the Sun, take it to be the size of the Earth's globe, and a circle around the Earth's surface. Then, the lunar (horizontal) parallax amounts to the difference in angular position, relative to the background of distant stars, of the Moon as seen from two different viewing positions on the Earth: one of the viewing positions is the place from which the Moon can be seen directly overhead at a given moment (that is, viewed along the vertical line in the diagram); and the other viewing position is a place from which the Moon can be seen on the horizon at the same moment (that is, viewed along one of the diagonal lines, from an Earth-surface position corresponding roughly to one of the blue dots on the modified diagram). The lunar (horizontal) parallax can alternatively be defined as the angle subtended at the distance of the Moon by the radius of the Earth—equal to angle p in the diagram when scaled-down and modified as mentioned above. The lunar horizontal parallax at any time depends on the linear distance of the Moon from the Earth. The Earth-Moon linear distance varies continuously as the Moon follows its perturbed and approximately elliptical orbit around the Earth. The range of the variation in linear distance is from about 56 to 63.7 Earth radii, corresponding to a horizontal parallax of about a degree of arc, but ranging from about 61.4' to about 54'. The '' Astronomical Almanac'' and similar publications tabulate the lunar horizontal parallax and/or the linear distance of the Moon from the Earth on a periodical e.g. daily basis for the convenience of astronomers (and of celestial navigators), and the study of how this coordinate varies with time forms part of
lunar theory Lunar theory attempts to account for the motions of the Moon. There are many small variations (or perturbations) in the Moon's motion, and many attempts have been made to account for them. After centuries of being problematic, lunar motion can now ...
. Parallax can also be used to determine the distance to the Moon. One way to determine the lunar parallax from one location is by using a lunar eclipse. A full shadow of the Earth on the Moon has an apparent radius of curvature equal to the difference between the apparent radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75 degrees, from which (with the solar apparent radius of 0.25 degrees) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60.27 Earth radii or This procedure was first used by Aristarchus of Samos and
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the e ...
, and later found its way into the work of
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
. The diagram at the right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the center of the planetary system and does not rotate. It also illustrates the important point that parallax need not be caused by any motion of the observer, contrary to some definitions of parallax that say it is, but may arise purely from motion of the observed. Another method is to take two pictures of the Moon at the same time from two locations on Earth and compare the positions of the Moon relative to the stars. Using the orientation of the Earth, those two position measurements, and the distance between the two locations on the Earth, the distance to the Moon can be triangulated: :\mathrm_ = \frac This is the method referred to by
Jules Verne Jules Gabriel Verne (;''Longman Pronunciation Dictionary''. ; 8 February 1828 – 24 March 1905) was a French novelist, poet, and playwright. His collaboration with the publisher Pierre-Jules Hetzel led to the creation of the '' Voyages extra ...
in '' From the Earth to the Moon'':
Until then, many people had no idea how one could calculate the distance separating the Moon from the Earth. The circumstance was exploited to teach them that this distance was obtained by measuring the parallax of the Moon. If the word parallax appeared to amaze them, they were told that it was the angle subtended by two straight lines running from both ends of the Earth's radius to the Moon. If they had doubts about the perfection of this method, they were immediately shown that not only did this mean distance amount to a whole two hundred thirty-four thousand three hundred and forty-seven miles (94,330 leagues) but also that the astronomers were not in error by more than seventy miles (≈ 30 leagues).


Solar parallax

After
Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulat ...
proposed his
heliocentric system Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth at ...
, with the Earth in revolution around the Sun, it was possible to build a model of the whole Solar System without scale. To ascertain the scale, it is necessary only to measure one distance within the Solar System, e.g., the mean distance from the Earth to the Sun (now called an
astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
, or AU). When found by triangulation, this is referred to as the ''solar parallax'', the difference in position of the Sun as seen from the Earth's center and a point one Earth radius away, i.e., the angle subtended at the Sun by the Earth's mean radius. Knowing the solar parallax and the mean Earth radius allows one to calculate the AU, the first, small step on the long road of establishing the size and expansion age of the visible Universe. A primitive way to determine the distance to the Sun in terms of the distance to the Moon was already proposed by Aristarchus of Samos in his book '' On the Sizes and Distances of the Sun and Moon''. He noted that the Sun, Moon, and Earth form a right triangle (with the right angle at the Moon) at the moment of first or last quarter moon. He then estimated that the Moon–Earth–Sun angle was 87°. Using correct
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
but inaccurate observational data, Aristarchus concluded that the Sun was slightly less than 20 times farther away than the Moon. The true value of this angle is close to 89° 50', and the Sun is about 390 times farther away. He pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the Sun was around 20 times larger than the Moon; this conclusion, although incorrect, follows logically from his incorrect data. It does suggest that the Sun is larger than the Earth, which could be taken to support the heliocentric model. Although Aristarchus' results were incorrect due to observational errors, they were based on correct geometric principles of parallax, and became the basis for estimates of the size of the Solar System for almost 2000 years, until the
transit of Venus frameless, upright=0.5 A transit of Venus across the Sun takes place when the planet Venus passes directly between the Sun and a superior planet, becoming visible against (and hence obscuring a small portion of) the solar disk. During a tr ...
was correctly observed in 1761 and 1769. This method was proposed by Edmond Halley in 1716, although he did not live to see the results. The use of Venus transits was less successful than had been hoped due to the
black drop effect The black drop effect is an optical phenomenon visible during a transit of Venus and, to a lesser extent, a transit of Mercury. Description Just after second contact, and again just before third contact during the transit, a small black "tear ...
, but the resulting estimate, 153 million kilometers, is just 2% above the currently accepted value, 149.6 million kilometers. Much later, the Solar System was "scaled" using the parallax of
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
s, some of which, such as Eros, pass much closer to Earth than Venus. In a favorable opposition, Eros can approach the Earth to within 22  a millikilometersres. During the opposition of 1900–1901, a worldwide program was launched to make parallax measurements of Eros to determine the
solar parallax Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby object ...
(or distance to the Sun), with the results published in 1910 by
Arthur Hinks Arthur Robert Hinks, CBE, FRS (26 May 1873 – 14 April 1945) was a British astronomer and geographer. As an astronomer, he is best known for his work in determining the distance from the Sun to the Earth (the astronomical unit) from 1900 to ...
of
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and
Charles D. Perrine Charles Dillon Perrine (July 28, 1867June 21, 1951) was an American astronomer at the Lick Observatory in California (1893-1909) who moved to Cordoba, Argentina to accept the position of Director of the Argentine National Observatory (1909-1936 ...
of the Lick Observatory,
University of California The University of California (UC) is a public land-grant research university system in the U.S. state of California. The system is composed of the campuses at Berkeley, Davis, University of California, Irvine, Irvine, University of Califor ...
. Perrine published progress reports in 1906 and 1908. He took 965 photographs with the Crossley Reflector and selected 525 for measurement. A similar program was then carried out, during a closer approach, in 1930–1931 by Harold Spencer Jones. The value of the
Astronomical Unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
(roughly the Earth-Sun distance) obtained by this program was considered definitive until 1968, when
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
and dynamical parallax methods started producing more precise measurements. Also
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, Marine radar, ships, spacecraft, guided missiles, motor v ...
reflections, both off Venus (1958) and off asteroids, like
Icarus In Greek mythology, Icarus (; grc, Ἴκαρος, Íkaros, ) was the son of the master craftsman Daedalus, the architect of the labyrinth of Crete. After Theseus, king of Athens and enemy of Minos, escaped from the labyrinth, King Minos sus ...
, have been used for solar parallax determination. Today, use of
spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, ...
telemetry Telemetry is the in situ collection of measurements or other data at remote points and their automatic transmission to receiving equipment (telecommunication) for monitoring. The word is derived from the Greek roots ''tele'', "remote", an ...
links has solved this old problem. The currently accepted value of solar parallax is 8".794 143.


Moving-cluster parallax

The open stellar cluster Hyades in Taurus extends over such a large part of the sky, 20 degrees, that the proper motions as derived from
astrometry Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, the Milky Way. Hist ...
appear to converge with some precision to a perspective point north of Orion. Combining the observed apparent (angular) proper motion in seconds of arc with the also observed true (absolute) receding motion as witnessed by the Doppler redshift of the stellar spectral lines, allows estimation of the distance to the cluster (151 light-years) and its member stars in much the same way as using annual parallax.


Dynamical parallax

Dynamical parallax has sometimes also been used to determine the distance to a supernova when the optical wavefront of the outburst is seen to propagate through the surrounding dust clouds at an apparent angular velocity, while its true propagation velocity is known to be the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
.


Derivation

For a right triangle, : \tan p = \frac , where p is the parallax, is approximately the average distance from the Sun to Earth, and d is the distance to the star. Using small-angle approximations (valid when the angle is small compared to 1
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
), : \tan x \approx x\text = x \cdot \frac \text = x \cdot 180 \cdot \frac \text , so the parallax, measured in arcseconds, is :p'' \approx \frac \cdot 180 \cdot \frac . If the parallax is 1", then the distance is :d = 1 \text \cdot 180 \cdot \frac \approx 206,265 \text \approx 3.2616 \text \equiv 1 \text . This ''defines'' the parsec, a convenient unit for measuring distance using parallax. Therefore, the distance, measured in parsecs, is simply d = 1 / p, when the parallax is given in arcseconds.


Error

Precise parallax measurements of distance have an associated error. This error in the measured parallax angle does not translate directly into an error for the distance, except for relatively small errors. The reason for this is that an error toward a smaller angle results in a greater error in distance than an error toward a larger angle. However, an approximation of the distance error can be computed by :\delta d = \delta \left( \right) =\left, \left( \right) \ \delta p = where ''d'' is the distance and ''p'' is the parallax. The approximation is far more accurate for parallax errors that are small relative to the parallax than for relatively large errors. For meaningful results in
stellar astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
, Dutch astronomer Floor van Leeuwen recommends that the parallax error be no more than 10% of the total parallax when computing this error estimate.


Spatio-temporal parallax

From enhanced ''relativistic positioning systems'', Spatio-temporal parallax generalizing the usual notion of parallax in space only has been developed. Then, event fields in spacetime can be deduced directly without intermediate models of light bending by massive bodies such as the one used in the PPN formalism for instance.


Metrology

Measurements made by viewing the position of some marker relative to something to be measured are subject to parallax error if the marker is some distance away from the object of measurement and not viewed from the correct position. For example, if measuring the distance between two ticks on a line with a ruler marked on its top surface, the thickness of the ruler will separate its markings from the ticks. If viewed from a position not exactly perpendicular to the ruler, the apparent position will shift and the reading will be less accurate than the ruler is capable of. A similar error occurs when reading the position of a pointer against a scale in an instrument such as an analog
multimeter A multimeter is a measuring instrument that can measure multiple electrical properties. A typical multimeter can measure voltage, resistance, and current, in which case it is also known as a volt-ohm-milliammeter (VOM), as the unit is equipped w ...
. To help the user avoid this problem, the scale is sometimes printed above a narrow strip of
mirror A mirror or looking glass is an object that reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the direction of the im ...
, and the user's eye is positioned so that the pointer obscures its reflection, guaranteeing that the user's line of sight is perpendicular to the mirror and therefore to the scale. The same effect alters the speed read on a car's speedometer by a driver in front of it and a passenger off to the side, values read from a
graticule Graticule may refer to: * An oscilloscope graticule scale * The reticle pattern in an optical instrument * Graticule (cartography), a grid of lines on a map See also * Grid (disambiguation) Grid, The Grid, or GRID may refer to: Common us ...
, not in actual contact with the display on an oscilloscope, etc.


Photogrammetry

When viewed through a stereo viewer, aerial picture pair offers a pronounced stereo effect of landscape and buildings. High buildings appear to "keel over" in the direction away from the center of the photograph. Measurements of this parallax are used to deduce the height of the buildings, provided that flying height and baseline distances are known. This is a key component of the process of photogrammetry.


Photography

Parallax error can be seen when taking photos with many types of cameras, such as twin-lens reflex cameras and those including viewfinders (such as rangefinder cameras). In such cameras, the eye sees the subject through different optics (the viewfinder, or a second lens) than the one through which the photo is taken. As the viewfinder is often found above the lens of the camera, photos with parallax error are often slightly lower than intended, the classic example being the image of a person with their head cropped off. This problem is addressed in
single-lens reflex camera A single-lens reflex camera (SLR) is a camera that typically uses a mirror and prism system (hence "reflex" from the mirror's reflection) that permits the photographer to view through the lens and see exactly what will be captured. With twin l ...
s, in which the viewfinder sees through the same lens through which the photo is taken (with the aid of a movable mirror), thus avoiding parallax error. Parallax is also an issue in image stitching, such as for panoramas.


Weapon sights

Parallax affects sighting devices of ranged weapons in many ways. On sights fitted on
small arms A firearm is any type of gun designed to be readily carried and used by an individual. The term is legally defined further in different countries (see Legal definitions). The first firearms originated in 10th-century China, when bamboo tubes ...
and bows, etc., the perpendicular distance between the sight and the weapon's launch axis (e.g. the bore axis of a gun)—generally referred to as "''sight height''"—can induce significant aiming errors when shooting at close range, particularly when shooting at small targets. This parallax error is compensated for (when needed) via calculations that also take in other variables such as bullet drop, windage, and the distance at which the target is expected to be. Sight height can be used to advantage when "sighting in" rifles for field use. A typical hunting rifle (.222 with telescopic sights) sighted in at 75m will still be useful from without needing further adjustment.


Optical sights

In some reticled optical instruments such as
telescope A telescope is a device used to observe distant objects by their emission, absorption, or reflection of electromagnetic radiation. Originally meaning only an optical instrument using lenses, curved mirrors, or a combination of both to obse ...
s,
microscope A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisi ...
s or in
telescopic sight A telescopic sight, commonly called a scope informally, is an optical sighting device based on a refracting telescope. It is equipped with some form of a referencing pattern – known as a '' reticle'' – mounted in a focally appropriate ...
s ("scopes") used on small arms and
theodolite A theodolite () is a precision optical instrument for measuring angles between designated visible points in the horizontal and vertical planes. The traditional use has been for land surveying, but it is also used extensively for building and ...
s, parallax can create problems when the reticle is not coincident with the
focal plane In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' ...
of the target image. This is because when the reticle and the target are not at the same focus, the optically corresponded distances being projected through the eyepiece are also different, and the user's eye will register the difference in parallaxes between the reticle and the target (whenever eye position changes) as a relative displacement on top of each other. The term ''parallax shift'' refers to the resultant apparent "floating" movements of the reticle over the target image when the user moves his/her head/eye laterally (up/down or left/right) behind the sight, i.e. an error where the reticle does not stay aligned with the user's
optical axis An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight. The optical axis is an imaginary line that defines the path along which light pro ...
. Some firearm scopes are equipped with a parallax compensation mechanism, which consists of a movable optical element that enables the optical system to shift the focus of the target image at varying distances into the same optical plane of the reticle (or vice versa). Many low-tier telescopic sights may have no parallax compensation because in practice they can still perform very acceptably without eliminating parallax shift. In this case, the scope is often set fixed at a designated parallax-free distance that best suits their intended usage. Typical standard factory parallax-free distances for hunting scopes are 100  yd (or 90 m) to make them suited for hunting shots that rarely exceed 300  yd/m. Some competition and military-style scopes without parallax compensation may be adjusted to be parallax free at ranges up to 300  yd/m to make them better suited for aiming at longer ranges. Scopes for guns with shorter practical ranges, such as
airguns An air gun or airgun is a gun that fires projectiles pneumatically with compressed air or other gases that are mechanically pressurized ''without'' involving any chemical reactions, in contrast to a firearm, which pressurizes gases ''chemical ...
,
rimfire rifle Rimfire ammunition is a type of firearm metallic cartridge whose primer is located within a hollow circumferential rim protruding from the base of its casing. When fired, the gun's firing pin will strike and crush the rim against the edge of ...
s,
shotgun A shotgun (also known as a scattergun, or historically as a fowling piece) is a long-barreled firearm designed to shoot a straight-walled cartridge known as a shotshell, which usually discharges numerous small pellet-like spherical sub- pr ...
s, and muzzleloaders, will have parallax settings for shorter distances, commonly for rimfire scopes and for shotguns and muzzleloaders. Airgun scopes are very often found with adjustable parallax, usually in the form of an adjustable objective (or "AO" for short) design, and may adjust down to as near as . Non-magnifying reflector or "reflex" sights can be theoretically "parallax free." But since these sights use parallel
collimated light A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A perfectly collimated light beam, with no divergence, would not disperse with distance. However, diffraction ...
this is only true when the target is at infinity. At finite distances, eye movement perpendicular to the device will cause parallax movement in the reticle image in exact relationship to the eye position in the cylindrical column of light created by the collimating optics. Firearm sights, such as some
red dot sights A red dot sight is a common classification for a type of non- magnifying reflector (or reflex) sight for firearms, and other devices that require aiming, that gives the user a point of aim in the form of an illuminated red dot. A standard desig ...
, try to correct for this via not focusing the reticle at infinity, but instead at some finite distance, a designed target range where the reticle will show very little movement due to parallax. Some manufacturers market reflector sight models they call "parallax free," but this refers to an optical system that compensates for off axis spherical aberration, an optical error induced by the spherical mirror used in the sight that can cause the reticle position to diverge off the sight's
optical axis An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight. The optical axis is an imaginary line that defines the path along which light pro ...
with change in eye position.


Artillery gunfire

Because of the positioning of field or naval artillery guns, each one has a slightly different perspective of the target relative to the location of the
fire-control system A fire-control system (FCS) is a number of components working together, usually a gun data computer, a director, and radar, which is designed to assist a ranged weapon system to target, track, and hit a target. It performs the same task as a ...
itself. Therefore, when aiming its guns at the target, the fire control system must compensate for parallax in order to assure that
fire Fire is the rapid oxidation of a material (the fuel) in the exothermic chemical process of combustion, releasing heat, light, and various reaction products. At a certain point in the combustion reaction, called the ignition point, flames ...
from each gun converges on the target.


Rangefinders

A coincidence rangefinder or parallax rangefinder can be used to find distance to a target.


Art

Several of
Mark Renn Mark Dennis Tate Renn (1952–2019) was a British sculptor who created several works of public art, mainly in the English Midlands. Renn was born in 1952 and trained in Birmingham. Although primarily known for his sculpture, his first commissi ...
's sculptural works play with parallax, appearing abstract until viewed from a specific angle. One such sculpture is ''The Darwin Gate'' (pictured) in
Shrewsbury Shrewsbury ( , also ) is a market town, civil parish, and the county town of Shropshire, England, on the River Severn, north-west of London; at the 2021 census, it had a population of 76,782. The town's name can be pronounced as either 'Sh ...
, England, which from a certain angle appears to form a dome, according to Historic England, in "the form of a Saxon helmet with a Norman window... inspired by features of St Mary's Church which was attended by Charles Darwin as a boy".


As a metaphor

In a philosophic/geometric sense: an apparent change in the direction of an object, caused by a change in observational position that provides a new line of sight. The apparent displacement, or difference of position, of an object, as seen from two different stations, or points of view. In contemporary writing, parallax can also be the same story, or a similar story from approximately the same timeline, from one book, told from a different perspective in another book. The word and concept feature prominently in
James Joyce James Augustine Aloysius Joyce (2 February 1882 – 13 January 1941) was an Irish novelist, poet, and literary critic. He contributed to the Modernism, modernist avant-garde movement and is regarded as one of the most influential and important ...
's 1922 novel, '' Ulysses''. Orson Scott Card also used the term when referring to Ender's Shadow as compared to
Ender's Game ''Ender's Game'' is a 1985 military science fiction novel by American author Orson Scott Card. Set at an unspecified date in Earth's future, the novel presents an imperiled humankind after two conflicts with an insectoid alien species they ...
. The metaphor is invoked by Slovenian philosopher
Slavoj Žižek Slavoj Žižek (, ; ; born 21 March 1949) is a Slovenian philosopher, cultural theorist and public intellectual. He is international director of the Birkbeck Institute for the Humanities at the University of London, visiting professor at New ...
in his 2006 book '' The Parallax View'', borrowing the concept of "parallax view" from the Japanese philosopher and literary critic Kojin Karatani. Žižek notes,


See also

*
Binocular disparity Binocular disparity refers to the difference in image location of an object seen by the left and right eyes, resulting from the eyes’ horizontal separation (parallax). The brain uses binocular disparity to extract depth information from the two-di ...
* Lutz–Kelker bias * Parallax mapping, in computer graphics * Parallax scrolling, in computer graphics *
Refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomen ...
, a visually similar principle caused by water, etc. * Spectroscopic parallax * Triangulation, wherein a point is calculated given its angles from other known points *
Trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
* True range multilateration, wherein a point is calculated given its distances from other known points * Xallarap


Notes


References


Bibliography

* * . * .


External links


Instructions for having background images on a web page use parallax effects


* BBC's ttp://www.bbc.co.uk/science/space/universe/questions_and_ideas/astronomical_distances/#p00bf0l7 Sky at Nightprogram: Patrick Moore demonstrates Parallax using Cricket. (Requires RealPlayer) * Berkeley Center for Cosmological Physic
Parallax


on an educational website, including a quick estimate of distance based on parallax using eyes and a thumb only * {{Portal bar, Astronomy, Stars, Spaceflight, Outer space, Solar System Optics Vision Angle Astrometry Geometry in computer vision Trigonometry