Parallax

Parallax is an apparent 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. The term is derived from the Greek παράλλαξις (parallaxis), meaning "alteration". Nearby objects have a larger parallax than more distant objects when observed from different positions, so parallax can be used to determine distances.

Astronomers use the principle of parallax to measure distances to celestial objects including to the Moon, the Sun, and to stars beyond the Solar System. For example, the Hipparcos satellite took measurements for over 100,000 nearby stars. This provides a basis for other distance measurements in astronomy, the cosmic distance ladder. Here, the term "parallax" is the angle or semi-angle of inclination between two sight-lines to the star.

Parallax also affects optical instruments such as binoculars, microscopes, and twin-lens reflex cameras that view objects from slightly different angles. Many animals, including 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 range, and in some variations also altitude to a target.

A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use an older-style "needle" speedometer gauge. 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.

Visual perception

is visible above the streetlight. The reflection in the water shows a virtual image of the Sun and the streetlight. 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.|thumb|right]] 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.

Distance measurement in astronomy

Stellar parallax

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. By observing parallax, measuring angles and using geometry, one can determine the distance to various objects. When the object in question is a star, the effect is known as stellar parallax.

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-years) is defined as the distance for which the annual parallax is 1 arcsecond. Annual parallax is normally measured by observing the position of a star at different times of the year 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 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 approximately 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's geometry 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 in order 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 and the eighth sphere (the fixed stars).

In 1989, the satellite Hipparcos was launched primarily for obtaining parallaxes and proper motions of nearby stars, increasing the reach of the method tenfold. 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 the Milky Way Galaxy. The European Space Agency's Gaia mission, due to launch in 2012 and come online in 2013, will be able to measure parallax angles to an accuracy of 10 microarcseconds, thus mapping nearby stars (and potentially planets) up to a distance of tens of thousands of light-years from earth.

Computation

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,

=== 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, it can exceed 1 degree. -- 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 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 formerly, of navigators), and the study of the way in which this coordinate varies with time forms part of lunar theory.

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 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. The diagram at right shows how daily lunar parallax arises on the geocentric and geostatic planetary model in which the Earth is at the centre 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 exactly 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: :$\backslash mathrm\{distance\}\_\{\backslash textrm\{moon\}\}\; =\; \backslash frac\; \{\backslash mathrm\{distance\}\_\{\backslash mathrm\{observerbase\}\}\}\; \{\backslash tan\; (\backslash mathrm\{angle\})\}$

This is the method referred to by Jules Verne 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 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).

Solar parallax

After Copernicus proposed his heliocentric system, 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, 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 centre 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 (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 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 actually about 390 times farther away. 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.

Also radar reflections, both off Venus (1958) and off asteroids, like Icarus, have been used for solar parallax determination. Today, use of spacecraft telemetry links has solved this old problem. The currently accepted value of solar parallax is 8".794 143.

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. 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.

Derivation

For a right triangle, : $\backslash sin\; p\; =\; \backslash frac\; \{1\; AU\}\; \{d\}\; ,$ 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), : $\backslash sin\; x\; \backslash approx\; x\backslash textrm\{\backslash \; radians\}\; =\; x\; \backslash cdot\; \backslash frac\; \{180\}\; \{\backslash pi\}\; \backslash textrm\{\backslash \; degrees\}\; =\; x\; \backslash cdot\; 180\; \backslash cdot\; \backslash frac\; \{3600\}\; \{\backslash pi\}\; \backslash textrm\{\backslash \; arcseconds\}\; ,$ so the parallax, measured in arcseconds, is :$p\text{'}\text{'}\; \backslash approx\; \backslash frac\; \{1\; \backslash textrm\{\backslash \; AU\}\}\; \{d\}\; \backslash cdot\; 180\; \backslash cdot\; \backslash frac\{3600\}\; \{\backslash pi\}\; .$ If the parallax is 1", then the distance is :$d\; =\; 1\; \backslash textrm\{\backslash \; AU\}\; \backslash cdot\; 180\; \backslash cdot\; \backslash frac\; \{3600\}\; \{\backslash pi\}\; \backslash approx\; 206,265\; \backslash textrm\{\backslash \; AU\}\; \backslash approx\; 3.2616\; \backslash textrm\{\backslash \; ly\}\; \backslash equiv\; 1\; \backslash textrm\{\backslash \; parsec\}\; .$ 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.

===Parallax error=== Precise parallax measurements of distance have an associated error. However 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 :$\backslash delta\; d\; =\; \backslash delta\; \backslash left(\; \{1\; \backslash over\; p\}\; \backslash right)\; =\backslash left|\; \{\backslash partial\; \backslash over\; \backslash partial\; p\}\; \backslash left(\; \{1\; \backslash over\; p\}\; \backslash right)\; \backslash right|\; \backslash delta\; p\; =\{\backslash delta\; p\; \backslash over\; p^2\}$ 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 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. 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 (for example, in an analog-display multimeter.) 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. This guarantees that the user's line of sight is perpendicular to the mirror and therefore to the scale.

Parallax can cause a speedometer reading to appear different to car's passenger from what it does to the driver. This is called a parallax error – the apparent shifting of the reading on a measuring instrument due to a change of the observer's viewing angle.

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.

Parallax error in photography

setting. Because the viewfinder is on top of the lens and of the close proximity of the subject, goggles are fitted in front of the rangefinder and a dedicated viewfinder installed to compensate for parallax.]] 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 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.

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

In computer graphics

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. Some hardware had explicit support for such layers, such as the Super Nintendo Entertainment System. 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 (such as Nintendo DS) still often use parallax. Parallax-based graphics continue to be used for many online applications where the bandwidth required by three-dimensional graphics is excessive.

Parallax compensation for firearm and archery sights

When a telescope sight is fitted to small arms, the distance between the sighting mechanism and the weapon's bore can introduce significant errors when firing at close range, particularly when firing at small targets. This also applies to archery where the shooter frequently relies on a single pin at close range.

Most telescopic sights lack parallax compensation because they can perform very acceptably without refinement for parallax. Telescopic sights manufacturers adjust these scopes at a distance that best suits their intended usage. Typical standard factory parallax adjustment distances for hunting telescopic sights are 100 yd or 100 m to make them suited for hunting shots that rarely exceed 300 yd/m. Some target and military style telescopic sights 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 rimfires, shotguns, and muzzleloaders will have shorter parallax settings, commonly 50 yd/m for rimfire scopes and 100 yd/m for shotguns and muzzleloaders. Scopes for airguns are very often found with adjustable parallax, usually in the form of an adjustable objective, or AO. These may adjust down as far as 3 yards (2.74 m).

In Naval Gunfire

Owing to the positioning of gun turrets on a warship or in the field, each one has a slightly different perspective of the target relative to the location of the fire control system itself. Therefore, when aiming its guns at the target, the fire control system must compensate for parallax in order to assure that fire from each turret converges on the target.

Parallax rangefinders

A Coincidence rangefinder or Parallax rangefinder can be used to find distance to a target.

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. 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 work The Parallax View. Žižek borrowed the concept of "parallax view" from the Japanese philosopher and literary critic Kojin Karatani. "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 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."

Notes

See also

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

Disparity

Spectroscopic parallax

Trigonometry

Xallarap

References

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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%

BBC's Sky at Night programme: Patrick Moore demonstrates Parallax using Cricket. (Requires RealPlayer)

Berkely Center for Cosmological Physics Parallax

Parallax on an educational website, including a quick estimate of distance based on parallax using eyes and a thumb only

Category:Optics Category:Vision Category:Angle Category:Astrometry Category:Geometry in computer vision Category:Technical factors of astrology Category:Astrological aspects Category:Trigonometry Category:Parallax