Revision 23253 of "Parallax" on enwiki

'''Parallax''' ([[Greek language|Greek]]: ''παραλλαγή (parallagé)'' =  alteration) is the change of angular [[location|position]] of two
[[stationary point]]s relative to each other as seen by an observer,
due to the motion of said observer. Or more simply put, it is the
apparent shift of an object against a background due to a change in
observer position.

==Informal introduction==

 [[Image:Parallax_Example.png|frame|right|Figure 1: A simplified example of parallax.]]

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 in front of the blue square.  When the 
viewpoint is changed to '''Viewpoint B''', the object ''appears'' to have moved to
in front of the red square.

==Use in distance measurement==
By [[observing]] parallax,
[[measurement|measuring]] [[angle]]s, and using [[geometry]]; one can
determine the [[distance]] to various objects. When this is in
reference to [[star]]s, the effect is known as '''stellar parallax'''.
The first measurements of a stellar parallax were made by
[[Friedrich Bessel]] in [[1838]].

Distance measurement by parallax is a special case of the principle of
[[triangulation]], where 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 only ''one'' side has been measured. Thus, the
careful measurement of the length of one baseline can fix the scale
of a triangulation network covering the whole nation. In parallax, the
triangle is extremely long and narrow, and by measuring both its
shortest side and the small top angle (the other two being close to
90 degrees), the long sides (in practice equal) can be determined.

==Parallax of the human eye==

With a nearby object in front of you, gaze at infinity. Cover one eye
with your hand. Then move your hand to cover your other eye instead. The
nearby object will seem to jump horizontally.

It is this effect that allows us — and certain other animals such as
cats — to see depth. It is used in simple stereo viewing devices,
such as the Viewmaster(TM) used to view stereoscopic scenery in the
form of two images taken from adjacent locations. The Apollo
astronauts on the Moon knew how to take such stereo pairs, clicking
two frames of the same object in locations shifted slightly
horizontally with respect to each other.

A way to allow a crowd of people simultaneously to view a stereoscopic
scene is to provide them with [[anaglyphic glasses]]. One glass is
red, the other green, and the stereo scene is produced by the printing
process in a corresponding fashion. It is generally believed that such
scenes are of necessity monochrome — red for the left image, green
for the right — but this is not quite true: working colour
anaglyphic scenes have been produced.

Instructions for self-producing anaglypic glasses by copying colour
onto an overhead projector sheet can be easily obtained. Better quality
glasses can also be purchased inexpensively from many science shops or
internet mail orders.

[[Autostereogram]]s exploit the effect of parallax to allow viewer to '''see''' 3D shapes in a single 2D image.

==Parallax and measurement instruments==

If an optical instrument — telescope, microscope, [[theodolite]] — is
imprecisely focused, the 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 carefully focus in order to 'eliminate the
parallax', and to check by moving one's head.

Also in non-optical measurements, e.g., the thickness of a ruler can
create parallax in fine measurements.  One is always cautioned in
science classes to "avoid parallax."  By this it is meant that one
should always 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]].  To help the user to
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.

In photography, one also talks about the parallax of a camera
viewfinder: for nearby objects, a viewfinder mounted on top of the
camera will show something different from what the lens 'sees', and
people's heads may be cut off. The problem does not exist for the
[[single lens reflex camera]], where the viewfinder looks (with the
aid of a movable mirror) through the same lens as is used for taking
the photograph.

==Photogrammetric parallax==

Aerial photograph pairs, when viewed through a stereo viewer, offer a
spectacular stereo effect of landscape and buildings. High buildings
appear to 'keel over' in the direction away from the centre of the
photograph.  Measuring this effect, also called parallax, allows one,
if the flying height and the distance between the aircraft's exposure
locations is known, to deduce the building's height.

==Lunar parallax==
[[Image:Lunarparallax_22_3_1988.png|thumb|right|Example of lunar parallax: Occultation of Pleiades by the Moon]]
[[Jules Verne]], ''[[From the Earth to the Moon]]'' ([[1865]]). "Up till 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)."

A primitive way to determine the lunar parallax from one location is
by using a lunar eclipse. The full shadow of the Earth on the Moon has
an apparent radius of curvature equal to the difference between the
apparent radii of the Earth and the Sun as seen from the Moon. This
radius can be seen to be equal to 0.75 degrees, from which (with the
solar apparent radius 0.25 degrees) we get an Earth apparent radius of
1 degree. This yields for the Earth-Moon distance 60 Earth radii or
384.000 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.
* distance<sub>moon</sub> = distance<sub>observer base</sub> / tan(angle)

==Solar parallax==

After [[Johannes Kepler]] discovered his [[Kepler's laws of planetary motion|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'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 &mdash; and thus the minimum age &mdash; of the visible Universe.

A primitive way of determining the distance to the Sun in terms of the distance to the Moon was already proposed by [[Aristarchus]]: if the Sun is relatively close by, the first and last quarters of the Moon will not happen in time precisely in the middle between new and full moon. Unfortunately the method (which unrealistically assumes regular circular motion for the Moon) becomes progressively imprecise for solar distances much larger than the distance of the Moon, and Aristachus obtained a nonsensical result. It is, however, in essence a parallax method.

[[Image:VenusTransitVermeer.png|600px|right|Measuring Venus transit times to determine solar parallax]]

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...


The 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 the parallax of
[[asteroid]]s, some of which, like [[433 Eros|Eros]], pass much
closer to Earth than Venus. 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]].

Also [[radar]] reflections, both off Venus (1958) and off asteroids, like
[[1566 Icarus|Icarus]], have been used for solar parallax
determination. Today, use of [[spacecraft]] [[telemetry]] links has solved
this old problem completely.

==Stellar parallax==
[[Image:Stellarparallax2.png|thumb|right|Stellar parallax motion]]

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 : 
:* The parallax <math>p'' = au/d*180*3600/\pi\,</math> in arc seconds
:** <math>au =</math> [[astronomical unit]] = Average distance from [[sun]] to [[earth]] = 1.4959e11 meters
:** <math>d =</math> distance to the star 
:* Picking a good unit of measure will cancel the constants.
:* Derivation:
:** '''(right triangle)''' <math>\sin p = au/d\,</math>
:** '''(small angle approximation)''' <math>\sin x ~= x\textrm{\ radians} = x*180/\pi \textrm{\ degrees} = x*180*3600/\pi</math> arcseconds 
:** parallax <math>p'' ~= au/d*180*3600/\pi</math>
:* If the parallax is 1", then the distance is <math>d = au*180*3600/\pi =</math> 206264 au = 3.2616 lyr = 1 parsec (This ''defines'' the parsec)
:* The parallax <math>p = 1/d</math> arc seconds, when distance 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; 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 [[star|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
purpose of obtaining parallaxes and [[proper motion|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
the first rungs on the cosmic scale ladder, out to a few hundred [[light year|light years]]. 
Beyond that, other methods must be taken into use: e.g.,
"[[spectroscopic]] parallaxes" &mdash; not really parallaxes at all. It is a
prototype of a "standard candle" method, where we observe the
apparent brightness of an object we know, based on some physical
theory, the true brightness of. For groups of stars we have the
[[Hertzsprung-Russell diagram]] which allows us to derive a star's
absolute brightness or [[magnitude]] <math>M</math> from its spectral type. The
observed (apparent) brightness or magnitude being <math>m</math>, we can then
derive its parallax <math>p</math> by

:<math>
 5 \log p + 5 = M - m,\,
</math>

called "spectroscopic parallax".

Further methods, mostly of the "standard candle" variety,
are the variable stars called [[Cepheid|Cepheids]] &mdash; the
absolute brightness of which depends on their observed period of
variation &mdash;, [[supernova]] brightnesses, [[globular cluster]] sizes and
brightnesses, complete [[galaxy]] brightnesses etc.  These are all much
more uncertain as they are not based on simple geometry. Yet,
parallaxes are the basis of everything, as they allow the calibration
of these more uncertain methods in the Solar neighbourhood.

A very modern method which is not a traditional parallax method but also geometric in nature, is "[[gravitational lens|gravitational lensing]] parallax". It depends on observing the differential time delay of brightness variations from a remote [[quasar]] reaching us by two different paths through the gravitational field or "lens" of a foreground galaxy.
If the redshifts of both the quasar and the foreground galaxy are known, one can show that the absolute distances of both are proportional to the differential delay, and can in fact be calculated given also the geometry of the gravitational lens on the celestial sphere.

All these independent techniques aim at determining [[Hubble's constant]], the constant describing how the [[redshift]] of galaxies, due to the Universe's expansion, depends on these galaxies' distance from us. Knowing Hubble's constant again allows us to determine, by simply running the film of the cosmic expansion backwards, how long ago it was when all these galaxies were collected in a single point -- the [[Big Bang]]. Current knowledge puts this at some 14.7 billion years ago, but with considerable uncertainty and dependence on various model assumptions.

== 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.

[[Category:Optics]]
[[Category:Computer vision]]
[[Category:Vision]]
[[Category:Physical quantity]]

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