Parallax
Definition
Parallax is the change of angular position of two\nstationary points relative to each other as seen by an observer,\ndue to the motion of said observer. Or more simply put, it is the\napparent shift of an object against a background due to a change in\nobserver position.\nUse in distance measurement
\nBy observing parallax,\nmeasuring angles, and using geometry; one can\ndetermine the distance to various objects. When this is in\nreference to stars, the effect is known as stellar parallax.\nThe first measurements of a stellar parallax were made by\nBessel, in 1838.
Distance measurement by parallax is a special case of the principle of\ntriangulation, where one can solve for all the sides and angles in\na network of triangles if, in addition to all the angles in the\nnetwork, the length of only one side has been measured. Thus, the\ncareful measurement of the length of one baseline can fix the scale\nof a triangulation network covering the whole nation. In parallax, the\ntriangle is extremely long and narrow, and by measuring both its\nshortest side and the small top angle (the other two being close to\n90 degrees), the long sides (in practice equal) can be determined.
Informal introduction
The two points mentioned in the definition above will be at different\ndistances from the observer. The visual effect of parallax is caused by\nthe fact that light follows straight lines. When the observer views\nthe nearer point, the line of his vision toward that point is at a\ngiven angle within the full arc of his vision. For example, let us\nsay that the view straight ahead is zero degrees, and one point,\nnearer the observer, is at minus five degrees while a point which is\nfarther away is at minus two degrees. The apparent angular distance\nbetween the points is a subjective three degrees to the viewer. If\nthe viewer moves ten meters to his right, the angular direction to the\nnearer object, as it is on a shorter radius, will change more than the\nangular direction to the farther object. So, for instance, when the\nangular direction to the nearer object is at minus ten degrees, the\nfather object may only have moved to minus three degrees. Now the\nsubjective angular difference in position is seven degrees. The\nobjects appear to have moved relative to each other.
It is because of this effect that, in a moving car, one can look at\ndistant mountains and see them seem to move (retard) in position\nbeneath a seemingly motionless moon. The moon is at such a distance\nthat the subjective angular change in position (direction) relative to an\nearth-bound observer is extremely slight, even as many miles are\ncovered. The mountains, much closer, exhibit a much greater apparent\nchange in angular position.
Put differently and somewhat more generally, distant objects appear to\nmove with the car. This can be explained as follows: looking out from\na car perpendicular to its motion, all objects move backward relative\nto the car, and for nearby objects the speed of change in direction is\nwhat the observer considers the normal consequence of his own\nmovement; however, for distant objects this backward change in\nabsolute direction is slow and much less obvious than the forward change in\ndirection relative to nearby objects. It seems as if distant objects\nmove parallel to the car with the same speed or only a little slower.
Parallax of the human eye
With a nearby object in front of you, gaze at infinity. Cover one eye\nwith your hand. Then move your hand to cover your other eye instead. The\nnearby object will seem to jump horizontally.
It is this effect that allows us -- and certain other animals such as\ncats -- to see depth. It is used in simple stereo viewing devices,\nsuch as the Viewmaster(TM) used to view stereoscopic scenery in the\nform of two images taken from adjacent locations. The Apollo\nastronauts on the Moon knew how to take such stereo pairs, clicking\ntwo frames of the same object in locations shifted slightly\nhorizontally with respect to each other.
A way to allow a crowd of people simultaneously to view a stereoscopic\nscene, is to provide them with anaglyphic glasses. One glass is\nred, the other green, and the stereo scene is produced by the printing\nprocess in a corresponding fashion. It is generally believed that such\nscenes are of necessity monochrome -- red for the left image, green\nfor the right --, but this is not quite true: working colour\nanaglyphic scenes have been produced.
Instructions for self-producing anaglypic glasses by copying colour\nonto an overhead projector sheet can be easily obtained. Better quality\nglasses can also be purchased inexpensively from many science shops or\ninternet mail orders.
\nParallax and measurement instruments
If an optical instrument -- telescope, microscope, theodolite -- is\nimprecisely focused, the cross-hairs will appear to move with respect\nto the object focused on if one moves one's head horizontally in front\nof the eyepiece. This is why it is important, especially when\nperforming measurements, to carefully focus in order to 'eliminate the\nparallax', and to check by moving one's head.
Also in non-optical measurements, e.g., the thickness of a ruler can\ncreate parallax in fine measurements. One is always cautioned in\nscience classes to "avoid parallax." By this it is meant that one\nshould always take measurements with one's eye on a line directly\nperpendicular to the ruler, so that the thickness of the ruler does\nnot create error in positioning for fine measurements. A similar\nerror can occur when reading the position of a pointer against a scale\nin an instrument such as a galvanometer. To help the user to\navoid this problem, the scale is sometimes printed above a narrow\nstrip of mirror, and the user positions his eye so that the\npointer obscures its own reflection. This guarantees that the user's\nline of sight is perpendicular to the mirror and therefore to the\nscale.
In photography, one also talks about the parallax of a camera\nviewfinder: for nearby objects, a viewfinder mounted on top of the\ncamera will show something different from what the lens 'sees', and\npeople's heads may be cut off. The problem does not exist for the\nsingle lens reflex camera, where the viewfinder looks (with the\naid of a movable mirror) through the same lens as is used for taking\nthe photograph.
\nPhotogrammetric parallax
Aerial photograph pairs, when viewed through a stereo viewer, offer a\nspectacular stereo effect of landscape and buildings. High buildings\nappear to 'keel over' in the direction away from the centre of the\nphotograph. Measuring this effect, also called parallax, allows one,\nif the flying height and the distance between the aircraft's exposure\nlocations is known, to deduce the building's height.
\nLunar parallax
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\nJules Verne, De la Terre à la Lune (1865). "Up till then, many people\nhad no idea how one could calculate the distance separating the Moon\nfrom the Earth. The circumstance was exploited to teach them that this\ndistance was obtained by measuring the parallax of the Moon. If the\nword parallax appeared to amaze them, they were told that it was the\nangle subtended by two straight lines running from both ends of the\nEarth's radius to the Moon. If they had doubts on the perfection of\nthis method, they were immediately shown that not only did this mean\ndistance amount to a whole two hundred thirty-four thousand three\nhundred and forty-seven miles (94.330 leagues), but also that the\nastronomers were not in error by more than sixty-six miles (-- 30\nleagues)."
A primitive way to determine the lunar parallax from one location is\nby using a lunar eclipse. The full shadow of the Earth on the Moon has\nan apparent radius of curvature equal to the difference between the\napparent radii of the Earth and the Sun as seen from the Moon. This\nradius can be seen to be equal to 0.75 degrees, from which (with the\nsolar apparent radius 0.25 degrees) we get an Earth apparent radius of\n1 degree. This yields for the Earth-Moon distance 60 Earth radii or\n384.000 km.
Another way to use parallax 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.\n* distancemoon = distanceobserver base / tan(angle)
Solar parallax
After Johannes Kepler discovered his Third Law, it was possible to build a scale model of the whole solar system, but without the scale. To fix the scale, it suffices to measure one distance within the solar system, e.g., the mean distance from the Earth to the Sun or astronomical unit (AU). When done 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 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 thus the minimum age -- of the visible Universe.
It was proposed by Edmund Halley in 1716, that the transit of Venus over the solar disc be used to derive the solar parallax. And so it was done in 1761 and 1769. There is the famous story of the French astronomer Guillaume Le Gentil, who travelled to India to observe the 1761 event, but didn't reach his destination in time due to war. He stayed on for the 1769 event, but then there were clouds blocking the Sun...
\nThe use of Venus transits was less successful than had been hoped due to the black drop effect. Much later, the solar system was 'scaled' using radar reflections from asteroids passing close to Earth (Eros, Icarus). Today, use of spacecraft telemetry links has solved this old problem completely.
Stellar parallax
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On an interstellar scale, parallax created by the different orbital positions of the Earth causes the stars to seem to move.
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 ends of the orbit, the parallax is half the maximum parallactic shift evident from the star viewed from the two points. The parsec is the distance for which the annual parallax is 1 arcsecond. A parsec equals 3.26 light years.
The distance of an object (in parsecs) can be computed as the reciprocal of the parallax. For instance, the nearest star, Alpha Centauri, has a parallax of 0.750". Therefore the distance is 1/0.750=1.33 parsecs or about 4.3 light years.
; Computation : \n:* The parallax in arc seconds\n:** astronomical unit = Average distance from sun to earth = 1.4959e11 meters\n:** distance to the star \n:* Picking a good unit of measure will cancel the constants.\n:* Derivation:\n:** (right triangle) \n:** (small angle approximation) arcseconds \n:** parallax \n:* If the parallax is 1", then the distance is 206264 au = 3.2616 lyr = 1 parsec (This defines the parsec)\n:* The parallax arc seconds, when distance given in parsecs\nThe 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 did not then occur to many people that the stars are so very much further away from us than the planets of the solar system as to render that argument useless.
Measurements of the annual parallax as the earth goes through its orbit was the first reliable way to determine the distances to the closest stars. This method was first used by Friedrich Wilhelm Bessel in 1838 when he measured the distance to 61 Cygni, and it remains the standard for calibrating other measurement methods (after the size of the orbit of the earth is measured by radar reflection on other planets). In 1989, a satellite called "Hipparcos" was launched with the main\npurpose of obtaining parallaxes and proper motions of nearby stars, increasing the reach of the method ten-fold.
Dynamic or moving-cluster parallax
The open stellar cluster 'Hyades' (Rain Stars) 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 us to estimate the distance of the cluster 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 was seen to propagate through the surrounding dust clouds at an apparent angular velocity, when we know its true propagation velocity to be that of light.
The scale of the Universe
All these various astronomical parallax methods allow us to establish\nthe first rungs on the cosmic scale ladder, out to a few hundred light\nyears. Beyond that, other methods must be taken into use: e.g.,\n"spectroscopic parallaxes" -- not really parallaxes at all. It is a\nprototype of a "standard candle" method, where we observe the\napparent brightness of an object we know, based on some physical\ntheory, the true brightness of. For groups of stars we have the\nHertzsprung-Russell diagram which allows us to derive a star's\nabsolute brightness or magnitude from its spectral type. The\nobserved (apparent) brightness or magnitude being , we can then\nderive its parallax by
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called "spectroscopic parallax".
Further methods, mostly of the "standard candle" variety,\nare the variable stars called Cepheids -- the\nabsolute brightness of which depends on their observed period of\nvariation --, supernova brightnesses, spherical cluster sizes and\nbrightnesses, complete galaxy brightnesses etc. These are all much\nmore uncertain as they are not based on simple geometry. Yet,\nparallaxes are the basis of everything, as they allow the calibration\nof these more uncertain methods in the Solar neighbourhood.
Parallax as a metaphor
In a philosophic/geometric sense: An apparent change in the direction\nof an object, caused by a change in observational position that\nprovides a new line of sight. The apparent displacement, or difference\nof position, of an object, as seen from two different stations, or\npoints of view.
Category:OpticsCategory:Computer vision
