Figure 1: A simplified example of parallax

Parallax, more accurately motion parallax, is the change of angular position of two observations of a single object relative to each other as seen by an observer, caused by the motion of the observer. In Physics, displacement is the vector that specifies the position of a point or a particle in reference to a previous position or to the origin of the chosen Simply put, it is the apparent shift of an object against the background that is caused by a change in the observer's position. The term is derived from the Greek παραλλαγή (parallagé), meaning "alteration".

Parallax is often thought of as the 'apparent motion' of an object against a distant background because of a perspective shift, as seen in Figure 1. When viewed from Viewpoint A, the object appears to be closer to the blue square. When the viewpoint is changed to Viewpoint B, the object appears to have moved in front of the red square.

In astronomy, parallax is the only direct method by which distances to objects beyond the Solar System can be measured. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study The Solar System consists of the Sun and those celestial objects bound to it by Gravity. The Hipparcos satellite has used the technique for over 100,000 nearby stars. Hipparcos (an Acronym for Hi gh P recision Par allax Co llecting S atellite) was an Astrometry mission A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth This provides the basis for all other distance measurements in astronomy, 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

Use in distance measurement

This image demonstrates parallax. The Sun is visible above the streetlight. The Sun (Sol is the Star at the center of the Solar System. The reflection in the water shows a virtual image of the Sun and the streetlight. In Optics, a virtual image is an image in which the outgoing rays from a point on the object never actually intersect at a point The location of the virtual image is below the surface of the water and thus simultaneously offers a different vantage point of the streetlight, which appears to be shifted relative to the stationary, background Sun.

By observing parallax, measuring angles and using geometry, one can determine the distance to various objects. Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Distance is a numerical description of how far apart objects are When the object in question is a star, the effect is known as stellar parallax. A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth The first successful measurements of stellar parallax were made by Friedrich Bessel in 1838, for the star 61 Cygni. Friedrich Wilhelm Bessel (22 July 1784 &ndash 17 March 1846 was a German Mathematician, Astronomer, and systematizer of the Bessel functions Not be confused with 16 Cygni, a more distant system containing two G-type stars harboring the Gas giant planet 16 Cygni Bb.

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. In Trigonometry and Geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either 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 (the other two being close to 90 degrees), the length of the long sides (in practice considered to be equal) can be determined.

Parallax error

Precise parallax measurements of distance usually have an associated error. The word error has different meanings and usages relative to how it is conceptually applied Thus a parallax may be described as some angle ± some angle-error. However this "± angle-error" does not translate directly into a ± error for the range, 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 means of the following:

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.

Parallax error in 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). A twin-lens reflex camera ( TLR) is a type of Camera with two objective lenses of the same Focal length. This article is about the definition for the short story by Raymond Carver, see Viewfinder (short story In Photography, a A rangefinder camera is a camera fitted with a Rangefinder: a range-finding focusing mechanism allowing the photographer to measure the subject distance and take photographs 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 person with his or her head cropped off. This problem is addressed in single-lens reflex cameras, 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. The single-lens reflex (SLR Camera uses an automatic moving mirror system which permits the photographer to see exactly what will be captured by the film or digital imaging

Binocular parallax

Instead of the observer's having to change position to experience parallax, he or she can simply close one and the other eye alternately. Eyes are organs that detect Light, and send signals along the Optic nerve to the visual areas of the brain Because the eyes are in different positions on the head, they yield the different views simultaneously. This is the basis of stereopsis. Stereopsis (from stereo meaning solidity and opsis meaning vision or Sight) is the process in Visual perception leading to the sensation

Parallax and measurement instruments

If an optical instrument — e. g. , a telescope, microscope, or theodolite — is imprecisely focused, its cross-hairs will appear to move with respect to the object focused on if one moves one's head horizontally in front of the eyepiece. A telescope is an instrument designed for the observation of remote objects and the collection of Electromagnetic radiation. A microscope ( Greek: ( micron) = small + ( skopein) = to look or see is an instrument for viewing objects that are A theodolite ( is an instrument for measuring both horizontal and vertical Angles as used in Triangulation networks This is why it is important, especially when performing measurements, to focus carefully in order to eliminate the parallax, and to check by moving one's head.

Also, in non-optical measurements the thickness of a ruler can create parallax in fine measurements. To avoid parallax error, one should take measurements with one's eye on a line directly perpendicular to the ruler so that the thickness of the ruler does not create error in positioning for fine measurements. A similar error can occur when reading the position of a pointer against a scale in an instrument such as a galvanometer. A galvanometer is a type of Ammeter; an instrument for detecting and measuring Electric current. To help the user avoid this problem, the scale is sometimes printed above a narrow strip of mirror, and the user positions his eye so that the pointer obscures its own reflection. A mirror is an object with a surface that has good Specular reflection; that is it is smooth enough to form an Image. Eyes are organs that detect Light, and send signals along the Optic nerve to the visual areas of the brain This guarantees that the user's line of sight is perpendicular to the mirror and therefore to the scale.

Photogrammetric parallax

Aerial picture pairs, when viewed through a stereo viewer, offer a pronounced stereo effect of landscape and buildings. High buildings appear to 'keel over' in the direction away from the centre 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 to the process of photogrammetry. Photogrammetry is the first Remote sensing technology ever developed in which geometric properties about objects are determined from photographic images

Lunar parallax

Example of lunar parallax: Occultation of Pleiades by the Moon

Lunar parallax involves the arithmetic of parallel stars between two inverted axes which rotate in angles that are only visible by the moon.

Jules Verne, in From the Earth to the Moon, wrote:

"Up till then, many people had no idea how one could calculate the distance separating the Moon from the Earth. Jules Gabriel Verne ( February 8 1828 &ndash March 24 1905) was a French Author who pioneered the science-fiction From the Earth to the Moon (De la Terre à la Lune 1865) is a humorous Science fantasy Novel by Jules Verne and is 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 on 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). "

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 degree, from which (with the solar apparent radius 0. 25 degree) we get an Earth apparent radius of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000 km. This procedure was first used by Aristarchus of Samos and Hipparchus, and later found its way into the work of Ptolemy. Aristarchus (Ἀρίσταρχος 310 BC - ca 230 BC) was a Greek Astronomer and Mathematician, born on the island of Hipparchus ( Greek; ca 190 BC &ndash ca 120 BC was a Greek Astronomer, Geographer, and Mathematician of the Hellenistic Claudius Ptolemaeus ( Greek: Klaúdios Ptolemaîos; after 83 &ndash ca

A more primitive way to determine the distance to the Moon would be to take two pictures of the Moon at exactly the same time from two locations on Earth, and compare the position of the Moon relative to the visible stars. Using the orientation of the Earth, and those two points, and a perpendicular displacement, a distance to the Moon can be triangulated.

Solar parallax

After Copernicus proposed his heliocentric system, with the Earth in revolution around the Sun, it was possible to build a scale model of the whole solar system, but without the 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, or AU). The astronomical unit ( AU or au or au or sometimes ua) is a unit of Length based on the distance from the Earth to the 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 centre and a point one Earth radius away, i. In Trigonometry and Geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either 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^[1] of the visible Universe. The age of the Universe is the time elapsed between the theory of the Big Bang and the present day

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. Aristarchus (Ἀρίσταρχος 310 BC - ca 230 BC) was a Greek Astronomer and Mathematician, born on the island of On the Sizes and Distances the Sun and Moon is the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310 BC He noted that the Sun, Moon, and Earth form a right triangle (right angle at the Moon) at the moment of first or last quarter moon. Lunar phase (or Moon phase refers to the appearance of the illuminated portion of the Moon as seen by an observer usually on Earth He then estimated that the Moon, Earth, Sun angle was 87°. Using correct geometry but inaccurate observational data, Aristarchus concluded that the Sun was slightly less than 20 times farther away than the Moon. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The true value of this angle is close to 89° 50', and the Sun is actually 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. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called 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 clearly larger than the Earth, which could be taken to support the heliocentric model. Although these 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 was correctly observed in 1761 and 1769. A transit of Venus across the Sun takes place when the Planet Venus passes directly between the Sun and Earth, obscuring a small portion of the

Measuring Venus transit times to determine solar parallax

This method was proposed by Edmond Halley in 1716 , although he did not live to see the results. Edmond Halley FRS (ˈɛdmənd ˈhɔːlɪ ( November 8, 1656 &ndash January 14, 1742) was an English Astronomer

The use of Venus transits was less successful than had been hoped due to the black drop effect, but the resulting estimate, 153 million kilometers, is just 2% over the currently accepted value, 149. The black drop effect is an optical phenomenon visible during a Transit of Venus and to a lesser extent a Transit of Mercury. 6 million kilometers.

Much later, the solar system was 'scaled' using the parallax of asteroids, some of which, like Eros, pass much closer to Earth than Venus. Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but TemplateInfobox Planet. --> 433 Eros (ˈɪərɒs irr'-os) is the first discovered Near-Earth asteroid In a favourable opposition, Eros can approach the Earth to within 22 million kilometres. Both the opposition of 1901 and that of 1930/1931 were used for this purpose, the calculations of the latter determination being completed by Astronomer Royal Sir Harold Spencer Jones. Sir Harold Spencer Jones KBE ( March 29 1890 Kensington, London &ndash November 3 1960) was an English

Also radar reflections, both off Venus (1958) and off asteroids, like Icarus, have been used for solar parallax determination. Radar is a system that uses electromagnetic waves to identify the range altitude direction or speed of both moving and fixed objects such as Aircraft, ships Today, use of spacecraft telemetry links has solved this old problem. A spacecraft is a Vehicle or machine designed for Spaceflight. Telemetry (synonymous with Telematics) is a Technology that allows the remote measurement and reporting of Information of interest to the system designer

Stellar parallax

Stellar parallax motion

On an interstellar scale, parallax created by the different orbital positions of the Earth causes nearby stars to appear to move relative to more distant stars. However, this effect is so small it is undetectable without extremely precise measurements.

The annual parallax is 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. Given two points on opposite sides of the orbit, the parallax is half the maximum parallactic shift evident from the star viewed from the two points. The parsec is defined as the distance for which the annual parallax is 1 arcsecond. History The first direct measurements of an object at interstellar distances were undertaken by German Astronomer Friedrich Wilhelm Bessel in 1838 A minute of arc, arcminute, or MOA is a unit of angular measurement, equal to one sixtieth (1/60 of one degree. A parsec equals 3. 26 light years.

The distance of an object (in parsecs) can be roughly computed as the reciprocal of the parallax. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which For instance, the Hipparcos satellite measured the parallax of the nearest star, Proxima Centauri, as 0. Hipparcos (an Acronym for Hi gh P recision Par allax Co llecting S atellite) was an Astrometry mission Proxima Centauri ( Latin la ''proximus proxima proximum'' meaning 'next to' or 'nearest to' is a Red dwarf Star approximately 4 77233 seconds of arc (±. 00242"). Therefore, the distance is 1/0. 772=1. 29 parsecs or about 4. 22 light years (±. 01 ly).

The angles involved in these calculations are very small. For example, 0. 772 arcseconds is approximately the angle subtended by an object about 2 centimeters in diameter (roughly the size of a U.S. quarter dollar) located about 5. In Geometry, an arc subtended by an Angle is a Curve whose endpoints are on the angle's two rays A quarter dollar is a coin worth 1/4 of a United States dollar, or 25 cents. 3 kilometers away.

Computation

The parallax in arc seconds

where

1 AU = 1 astronomical unit = Average distance from the Sun to earth = 1. The astronomical unit ( AU or au or au or sometimes ua) is a unit of Length based on the distance from the Earth to the The Sun (Sol is the Star at the center of the Solar System. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 4959 · 10¹¹ m

d = distance to the star

Picking a good unit of measure will cancel the constants. Derivation:

right triangle

small angle approximation

arcseconds

parallax

If the parallax is 1", then the distance is = 206,265 AU = 3. 2616 ly = 1 parsec (This defines the parsec)

The parallax arcseconds, when the distance is given in parsecs

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. History The first direct measurements of an object at interstellar distances were undertaken by German Astronomer Friedrich Wilhelm Bessel in 1838 In Astronomy, heliocentrism is the theory that the Sun is at the center of the Solar System. It is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away; but for various reasons such a gigantic size seemed entirely implausible. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

Measurement of the annual parallax as the earth goes through its orbit was the first reliable way to determine the distances to the closest stars. A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth This method was first successfully used by Friedrich Wilhelm Bessel in 1838 when he measured the distance to 61 Cygni with a heliometer, and it remains the standard for calibrating other measurement methods (after the size of the orbit of the earth is measured by radar reflection of other planets). Friedrich Wilhelm Bessel (22 July 1784 &ndash 17 March 1846 was a German Mathematician, Astronomer, and systematizer of the Bessel functions Not be confused with 16 Cygni, a more distant system containing two G-type stars harboring the Gas giant planet 16 Cygni Bb. Heliometer (from Greek sun and measure) is an instrument originally designed for measuring the variation of the Sun 's diameter at different Radar is a system that uses electromagnetic waves to identify the range altitude direction or speed of both moving and fixed objects such as Aircraft, ships

In 1989, the satellite Hipparcos was launched primarily for obtaining parallaxes and proper motions of nearby stars, increasing the reach of the method tenfold. Hipparcos (an Acronym for Hi gh P recision Par allax Co llecting S atellite) was an Astrometry mission The proper motion of a Star is the measurement of its change in position in the sky over time after Improper motions are accounted for Even so, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away — a little more than one percent of the diameter of our galaxy. The Milky Way (a translation of the Latin Via Lactea, in turn derived from the Greek Γαλαξίας (Galaxias sometimes referred to simply The European Space Agency's Gaia mission, due to launch in 2011 and come online in 2012, will be able to measure parallax angles to an accuracy of 1 micro-arc second, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light years from earth. The European Space Agency ( ESA) established in 1975 is an intergovernmental organisation dedicated to the exploration of space, currently with 17 member Gaia is a European Space Agency (ESA Astrometry space mission and a successor to the ESA Hipparcos mission ^[2]

Dynamic or 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 appear to converge with some precision to a perspective point north of Orion. The Hyades ( Greek Ὑάδες also known as Melotte 25 or Collinder 50 or Caldwell 41 is the nearest Open cluster to the Taurus (it looks like a bull (ˈtɔrəs bull, symbol, Unicode ♉ is one of the Constellations of the Zodiac. Astrometry is the branch of Astronomy that relates to precise measurements and explanations of the positions and movements of Stars and other celestial bodies 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.

Dynamic parallax has sometimes also been used to determine the distance to a supernova, when the optical wave front 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.

In computer graphics

Main article: Parallax scrolling

Main article: Parallax mapping

In many early graphical applications, such as video games, the scene was constructed of independent layers that were scrolled at different speeds when the player/cursor moved. Parallax scrolling is a special Scrolling technique in Computer graphics, seen first in the 1982 Arcade game Moon Patrol Parallax mapping (also called offset mapping or virtual displacement mapping) is an enhancement of the Bump mapping or Normal mapping techniques Some hardware had explicit support for such layers, such as the Super Nintendo Entertainment System. The Super Nintendo Entertainment System or Super NES (also called SNES and Super Nintendo) is a 16-bit Video game console that was This gave some layers the appearance of being farther away than others and was useful for creating an illusion of depth, but only worked when the player was moving. Now, most games are based on much more comprehensive three-dimensional graphic models, although portable game systems (DS, PSP) still often use parallax.

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 time line, from one book told from a different perspective in another book. The word and concept feature prominently in James Joyce's 1922 novel, Ulysses. James Augustine Aloysius Joyce (2 February 1882 &ndash 13 January 1941 was an Irish expatriate writer widely considered to be one of the most influential writers of the Ulysses is a novel by James Joyce, first serialized in parts in the American journal The Little Review from March 1918 to December 1920 Orson Scott Card also used the term when referring to Ender's Shadow as compared to Ender's Game. Orson Scott Card (born August 24, 1951) is a bestselling American Author, Critic, political writer and speaker. Ender's Shadow (1999 is a parallel Science fiction novel by Orson Scott Card with a plot covering some of the events in Ender's Game (1985 is one of the best-known novels by Orson Scott Card.

The metaphor is also invoked in the magnum opus of Slovenian philosopher Slavoj Žižek in his work 'The Parallax View'. Slavoj Žižek (ˈslavoj ˈʒiʒɛk (born 21 March 1949) is a Post-Marxist Sociologist, Philosopher, and Cultural critic "The philosophical twist to be added ((to parallax)), of course, is that the observed distance is not simply subjective, since the same object which exists 'out there' is seen from two different stances, or points of view. It is rather that, as Hegel would have put it, subject and object are inherently mediated so that an 'epistemological' shift in the subject's point of view always reflects an ontological shift in the object itself. Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge In Philosophy, ontology (from the Greek, genitive: of being (part Or -to put it in Lacanese- the subject's gaze is always-already inscribed into the perceived object itself, in the guise of its 'blind spot,' that which is 'in the object more than object itself', the point from which the object itself returns the gaze. Jacques-Marie-Émile Lacan (French ʒak lakɑ̃ ( April 13, 1901 &ndash September 9, 1981) was a French Psychoanalyst Sure the picture is in my eye, but I am also in the picture ^[3]

References

General references

• Zeilik, Michael A. & Gregory, Stephan A. (1998), Introductory Astronomy & Astrophysics (4th ed. ), Saunders College Publishing, pp. 43–44, ISBN 0030062284 

See also

• Disparity
• Triangulation, wherein a point is calculated given its angles from other known points
• Trilateration, wherein a point is calculated given its distances from other known points
• Trigonometry
• The Extragalactic Distance Scale

External links

• Instructions for having background images on a web page use parallax effects
• Actual parallax project measuring the distance to the moon within 2.3%
• Java applet demonstrating stellar parallax
• BBC's Sky at Night programme: Patrick Moore demonstrates Parallax using Cricket. In Trigonometry and Geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either Trilateration is a method of determining the relative positions of objects using the Geometry of Triangles in a similar fashion as Triangulation. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. (Requires RealPlayer)
• Definition of "parallax" at http://reference.dictionary.com
• "What is parallax?