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Parallax

(Redirected from Trigonometric parallax)
This article is about the apparent displacement of an object viewed from different positions. For other uses, see Parallax (disambiguation).

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 half-angle of inclination between those two lines.[1][2] Due to foreshortening, nearby objects show a larger parallax than farther objects, so parallax can be used to determine distances.

A simplified illustration of the parallax of an object against a distant background due to a perspective shift. When viewed from "Viewpoint A", the object appears to be in front of the blue square. When the viewpoint is changed to "Viewpoint B", the object appears to have moved in front of the red square.
This animation is an example of parallax. As the viewpoint moves side to side, the objects in the distance appear to move more slowly than the objects close to the camera. In this case, the white cube in front appears to move faster than the green cube in the middle of the far background.

To measure large distances, such as the distance of a planet or a star from Earth, 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.[a] These distances form the lowest rung of what is called "the cosmic distance ladder", 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, microscopes, 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 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

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Main articles: stereopsis, depth perception, binocular vision, and binocular disparity
 
In this photograph, the Sun is visible above the top of the streetlight. In the reflection on the water, the Sun appears in line with the streetlight because the virtual image is formed from a different viewing position.

Because 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.[3]

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.[4] The motion parallax is exploited also in wiggle stereoscopy, computer graphics that provide depth cues through viewpoint-shifting animation rather than through binocular vision.

Distance measurement

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Parallax theory for finding naval distances

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 angles, and using geometry, one can determine distance.

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,[5] leaving the other two close to 90  degrees), the length of the long sides (in practice considered to be equal) can be determined.

In astronomy, assuming the angle is small, the distance to a star (measured in parsecs) is the reciprocal of the parallax (measured in arcseconds):   For example, the distance to Proxima Centauri is 1/0.7687 = 1.3009 parsecs (4.243 ly).[6]

On Earth, a coincidence rangefinder or parallax rangefinder can be used to find distance to a target. In surveying, the problem of resection explores angular measurements from a known baseline for determining an unknown point's coordinates.

Astronomy

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This section is an excerpt from Parallax in astronomy.[edit]
 
Stellar parallax motion from annual parallax. Half the apex angle is the parallax angle.
 
Parallax is an angle subtended by a line on a point. In the upper diagram, the Earth in its orbit sweeps the parallax angle subtended on the Sun. The lower diagram shows an equal angle swept by the Sun in a geostatic model. A similar diagram can be drawn for a star except that the angle of parallax would be minuscule.

The most important fundamental distance measurements in astronomy come from trigonometric parallax, as applied in the stellar parallax method. As the Earth orbits the Sun, the position of nearby stars will appear to shift slightly against the more distant background. These shifts are angles in an isosceles triangle, with 2 AU (the distance between the extreme positions of Earth's orbit around the Sun) making the base leg of the triangle and the distance to the star being the long equal-length legs. The amount of shift is quite small, even for the nearest stars, measuring 1 arcsecond for an object at 1 parsec's distance (3.26 light-years), and thereafter decreasing in angular amount as the distance increases. Astronomers usually express distances in units of parsecs (parallax arcseconds); light-years are used in popular media.

Because parallax becomes smaller for a greater stellar distance, useful distances can be measured only for stars which are near enough to have a parallax larger than a few times the precision of the measurement. In the 1990s, for example, the Hipparcos mission obtained parallaxes for over a hundred thousand stars with a precision of about a milliarcsecond,[7] providing useful distances for stars out to a few hundred parsecs. The Hubble Space Telescope's Wide Field Camera 3 has the potential to provide a precision of 20 to 40 microarcseconds, enabling reliable distance measurements up to 5,000 parsecs (16,000 ly) for small numbers of stars.[8][9] The Gaia space mission provided similarly accurate distances to most stars brighter than 15th magnitude.[10]

Distances can be measured within 10% as far as the Galactic Center, about 30,000 light years away. Stars have a velocity relative to the Sun that causes proper motion (transverse across the sky) and radial velocity (motion toward or away from the Sun). The former is determined by plotting the changing position of the stars over many years, while the latter comes from measuring the Doppler shift of the star's spectrum caused by motion along the line of sight. For a group of stars with the same spectral class and a similar magnitude range, a mean parallax can be derived from statistical analysis of the proper motions relative to their radial velocities. This statistical parallax method is useful for measuring the distances of bright stars beyond 50 parsecs and giant variable stars, including Cepheids and the RR Lyrae variables.[11]

 
Parallax measurements may be an important clue to understanding three of the universe's most elusive components: dark matter, dark energy and neutrinos.[12]
 
Hubble Space Telescope precision stellar distance measurement has been extended 10 times further into the Milky Way.[13]

The motion of the Sun through space provides a longer baseline that will increase the accuracy of parallax measurements, known as secular parallax. For stars in the Milky Way disk, this corresponds to a mean baseline of 4 AU per year, while for halo stars the baseline is 40 AU per year. After several decades, the baseline can be orders of magnitude greater than the Earth–Sun baseline used for traditional parallax. However, secular parallax introduces a higher level of uncertainty because the relative velocity of observed stars is an additional unknown. When applied to samples of multiple stars, the uncertainty can be reduced; the uncertainty is inversely proportional to the square root of the sample size.[14]

Moving cluster parallax is a technique where the motions of individual stars in a nearby star cluster can be used to find the distance to the cluster. Only open clusters are near enough for this technique to be useful. In particular the distance obtained for the Hyades has historically been an important step in the distance ladder.

Other individual objects can have fundamental distance estimates made for them under special circumstances. If the expansion of a gas cloud, like a supernova remnant or planetary nebula, can be observed over time, then an expansion parallax distance to that cloud can be estimated. Those measurements however suffer from uncertainties in the deviation of the object from sphericity. Binary stars which are both visual and spectroscopic binaries also can have their distance estimated by similar means, and do not suffer from the above geometric uncertainty. The common characteristic to these methods is that a measurement of angular motion is combined with a measurement of the absolute velocity (usually obtained via the Doppler effect). The distance estimate comes from computing how far the object must be to make its observed absolute velocity appear with the observed angular motion.

Expansion parallaxes in particular can give fundamental distance estimates for objects that are very far, because supernova ejecta have large expansion velocities and large sizes (compared to stars). Further, they can be observed with radio interferometers which can measure very small angular motions. These combine to provide fundamental distance estimates to supernovae in other galaxies.[15] Though valuable, such cases are quite rare, so they serve as important consistency checks on the distance ladder rather than workhorse steps by themselves.

Metrology

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The correct line of sight needs to be used to avoid parallax error.

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. To help the user avoid this problem, the scale is sometimes printed above a narrow strip of mirror, 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, not in actual contact with the display on an oscilloscope, etc.

Photogrammetry

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Main article: 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

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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 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.

Parallax is also an issue in image stitching, such as for panoramas.

  • Contax III rangefinder camera with macro photography setting. Because the viewfinder is on top of the lens and near the subject, goggles are fitted in front of the rangefinder and a dedicated viewfinder is installed to compensate for parallax. 
    Contax III rangefinder camera with macro photography setting. Because the viewfinder is on top of the lens and near the subject, goggles are fitted in front of the rangefinder and a dedicated viewfinder is installed to compensate for parallax.
  • Failed panoramic image due to the parallax, since the axis of rotation of the tripod is not the same as the focal point. 
    Failed panoramic image due to the parallax, since the axis of rotation of the tripod is not the same as the focal point.

Weapon sights

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Parallax affects sighting devices of ranged weapons in many ways. On sights fitted on small arms 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.[16] 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.[17] 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 50 to 200 m (55 to 219 yd) without needing further adjustment.[citation needed]

Optical sights

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Further information: Telescopic sight § Parallax compensation
 
Simple animation demonstrating the effects of parallax compensation in telescopic sights, as the eye moves relative to the sight.

In some reticled optical instruments such as telescopes, microscopes or in telescopic sights ("scopes") used on small arms and theodolites, parallax can create problems when the reticle is not coincident with the focal plane 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,[18] i.e. an error where the reticle does not stay aligned with the user's optical axis.

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. [citation needed] Scopes for guns with shorter practical ranges, such as airguns, rimfire rifles, shotguns, and muzzleloaders, will have parallax settings for shorter distances, commonly 50 m (55 yd) for rimfire scopes and 100 m (110 yd) for shotguns and muzzleloaders. [citation needed] 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 3 metres (3.3 yd).[citation needed]

Non-magnifying reflector or "reflex" sights can be theoretically "parallax free". But since these sights use parallel collimated light 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.[19][20] Firearm sights, such as some red dot sights, 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.[19] Some manufacturers market reflector sight models they call "parallax free",[21] 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 with change in eye position.[22][23]

Artillery-fire

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Because of the positioning of field or naval artillery, each gun has a slightly different perspective of the target relative to the location of the fire-control system. When aiming guns at the target, the fire control system must compensate for parallax to assure that fire from each gun converges on the target.

Art

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Several of Mark Renn's sculptural works play with parallax, appearing abstract until viewed from a specific angle. One such sculpture is The Darwin Gate (pictured) in Shrewsbury, 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".[24]

their separate columns, each arched at the top 
an apparent dome 
Viewed from a certain angle the curves of the three separate columns of The Darwin Gate appear to form a dome

As a metaphor

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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's 1922 novel, Ulysses. Orson Scott Card also used the term when referring to Ender's Shadow as compared to Ender's Game.

The metaphor is invoked by Slovenian philosopher Slavoj Žižek in his 2006 book The Parallax View, borrowing the concept of "parallax view" from the Japanese philosopher and literary critic Kojin Karatani. Žižek notes

The philosophical twist to be added (to parallax), of course, is that the observed distance is not simply "subjective", since the same object that 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. 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 the object itself," the point from which the object itself returns the gaze. "Sure the picture is in my eye, but I am also in the picture"...[25]

— Slavoj Žižek, The Parallax View

See also

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Notes

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  1. ^ In the past diurnal parallax was also used to measure distances to celestial objects within the Solar System. This method has now been superseded by more accurate techniques.

References

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  1. ^ "Parallax". Shorter Oxford English Dictionary. 1968. Mutual inclination of two lines meeting in an angle
  2. ^ "Parallax". Oxford English Dictionary (Second ed.). 1989. Astron. Apparent displacement, or difference in the apparent position, of an object, caused by an actual change (or difference) of the position of the point of observation; spec. the angular amount of such displacement or difference of position, being the angle contained between the two straight lines drawn to the object from the two different points of view and constituting a measure of the distance of the object.
  3. ^ Steinman, Scott B.; Garzia, Ralph Philip (2000). Foundations of Binocular Vision: A Clinical perspective. McGraw-Hill Professional. pp. 2–5. ISBN 978-0-8385-2670-5.
  4. ^ Steinman & Garzia 2000, p. 180.
  5. ^ Zeilik & Gregory 1998, p. 44.
  6. ^ Benedict, G. Fritz, et al. (1999). "Interferometric Astrometry of Proxima Centauri and Barnard's Star Using Hubble Space Telescope Fine Guidance Sensor 3: Detection Limits for Substellar Companions". The Astronomical Journal. 118 (2): 1086–1100. arXiv:Astro-ph/9905318. Bibcode:1999AJ....118.1086B. doi:10.1086/300975. S2CID 18099356.
  7. ^ Perryman, M. A. C.; et al. (1999). "The HIPPARCOS Catalogue". Astronomy and Astrophysics. 323: L49–L52. Bibcode:1997A&A...323L..49P.
  8. ^ Harrington, J. D.; Villard, R. (10 April 2014). "NASA's Hubble Extends Stellar Tape Measure 10 Times Farther Into Space". NASA. Archived from the original on 17 February 2019. Retrieved 17 October 2014.
  9. ^ Riess, A. G.; Casertano, S.; Anderson, J.; MacKenty, J.; Filippenko, A. V. (2014). "Parallax Beyond a Kiloparsec from Spatially Scanning the Wide Field Camera 3 on the Hubble Space Telescope". The Astrophysical Journal. 785 (2): 161. arXiv:1401.0484. Bibcode:2014ApJ...785..161R. doi:10.1088/0004-637X/785/2/161. S2CID 55928992.
  10. ^ Brown, A. G. A.; et al. (Gaia collaboration) (August 2018). "Gaia Data Release 2: Summary of the contents and survey properties". Astronomy & Astrophysics. 616. A1. arXiv:1804.09365. Bibcode:2018A&A...616A...1G. doi:10.1051/0004-6361/201833051.
  11. ^ B., Baidyanath (2003). An Introduction to Astrophysics. PHI Learning Private Limited. ISBN 978-81-203-1121-3.
  12. ^ "Hubble finds Universe may be expanding faster than expected". Archived from the original on 11 September 2018. Retrieved 3 June 2016.
  13. ^ "Hubble stretches the stellar tape measure ten times further". ESA/Hubble Images. Archived from the original on October 30, 2017. Retrieved April 12, 2014.
  14. ^ Popowski, P.; Gould, A. (1998). "Mathematics of Statistical Parallax and the Local Distance Scale". arXiv:astro-ph/9703140.
  15. ^ Bartel, N.; et al. (1994). "The shape, expansion rate and distance of supernova 1993J from VLBI measurements". Nature. 368 (6472): 610–613. Bibcode:1994Natur.368..610B. doi:10.1038/368610a0. S2CID 4316734.
  16. ^ "Ballistic Explorer Help". www.dexadine.com. Archived from the original on 2011-09-28.
  17. ^ "Crossbows / Arrows & Bolts / Trajectory / Trajectories". www.crossbowmen.com. Archived from the original on 2011-07-08.
  18. ^ "Setting Up An Air Rifle And Telescopic Sight For Field Target – An Instruction Manual For Beginners, page 16". Retrieved 2019-10-28.
  19. ^ a b "Encyclopedia of Bullseye Pistol". www.bullseyepistol.com. Archived from the original on 2011-07-08.
  20. ^ John P. Butler (1944). "The Reflector Sight". American Rifleman. National Rifle Association. p. 31.
  21. ^ AFMOTGN (24 July 2008). "Aimpoint's parallax-free, double lens system... AFMO.com". Archived from the original on 2 July 2016 – via YouTube.
  22. ^ AR15.COM. "How Aimpoints, EOTech, And Other Parallax-Free Optics Work – AR15.COM". www.ar15.com.{{cite web}}: CS1 maint: numeric names: authors list (link)
  23. ^ "Gunsight – Patent 5901452 – general description of a mManginmirror system". Archived from the original on 2012-10-07.
  24. ^ Historic England. "Darwin Gate (1490992)". Research records (formerly PastScape). Retrieved 4 January 2020.
  25. ^ Žižek, Slavoj (2006). The Parallax View. The MIT Press. pp. 17. ISBN 978-0-262-24051-2.

Bibliography

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