Parallax (from Ancient Greek παράλλαξις
(parallaxis), meaning 'alternation') 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 semi-angle of inclination between those
two lines. Due to foreshortening, nearby objects
show a larger parallax than farther objects
when observed from different positions, so
parallax can be used to determine distances.
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. 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,
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 a needle-style 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 ==
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.The motion
parallax is exploited also in wiggle stereoscopy,
computer graphics which provide depth cues
through viewpoint-shifting animation rather
than through binocular vision.
== Astronomy ==
Parallax arises due to change in viewpoint
occurring due to motion of the observer, of
the observed, or of both. What is essential
is relative motion. By observing parallax,
measuring angles, and using geometry, one
can determine distance. Astronomers also use
the word "parallax" as a synonym for "distance
measurement" by other methods: see parallax
(disambiguation)#Astronomy.
=== Stellar parallax ===
Stellar parallax created by the relative motion
between the Earth and a star can be seen,
in the Copernican model, as arising from the
orbit of the Earth around the Sun: the star
only appears to move relative to more distant
objects in the sky. In a geostatic model,
the movement of the star would have to be
taken as real with the star oscillating across
the sky with respect to the background stars.
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 (then the most distant known planet)
and the eighth sphere (the fixed stars).In
1989, the satellite Hipparcos was launched
primarily for obtaining improved parallaxes
and proper motions for over 100,000 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,
launched in December 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.
In April 2014, NASA astronomers reported that
the Hubble Space Telescope, by using spatial
scanning, can now precisely measure distances
up to 10,000 light-years away, a ten-fold
improvement over earlier measurements.
=== Distance measurement ===
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, leaving the other two
close to 90 degrees), the length of the long
sides (in practice considered to be equal)
can be determined.
Assuming the angle is small (see derivation
below), the distance to an object (measured
in parsecs) is the reciprocal of the parallax
(measured in arcseconds):
d
(
p
c
)
=
1
/
p
(
a
r
c
s
e
c
)
.
{\displaystyle d(\mathrm {pc} )=1/p(\mathrm
{arcsec} ).}
For example, the distance to Proxima Centauri
is 1/0.7687=1.3009 parsecs (4.243 ly).
=== Diurnal parallax ===
Diurnal parallax is a parallax that varies
with rotation of the Earth or with difference
of location on the Earth. The Moon and to
a smaller extent the terrestrial planets or
asteroids seen from different viewing positions
on the Earth (at one given moment) can appear
differently placed against the background
of fixed stars.
=== 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.The
diagram (above) for stellar parallax can illustrate
lunar parallax as well, if the diagram is
taken to be scaled right down and slightly
modified. Instead of 'near star', read 'Moon',
and instead of taking the circle at the bottom
of the diagram to represent the size of the
Earth's orbit around the Sun, take it to be
the size of the Earth's globe, and of a circle
around the Earth's surface. Then, the lunar
(horizontal) parallax amounts to the difference
in angular position, relative to the background
of distant stars, of the Moon as seen from
two different viewing positions on the Earth:
one of the viewing positions is the place
from which the Moon can be seen directly overhead
at a given moment (that is, viewed along the
vertical line in the diagram); and the other
viewing position is a place from which the
Moon can be seen on the horizon at the same
moment (that is, viewed along one of the diagonal
lines, from an Earth-surface position corresponding
roughly to one of the blue dots on the modified
diagram).
The lunar (horizontal) parallax can alternatively
be defined as the angle subtended at the distance
of the Moon by the radius of the Earth—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 of celestial 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 60.27 Earth radii or 384,399 kilometres
(238,854 mi) This procedure was first used
by Aristarchus of Samos and Hipparchus, and
later found its way into the work of Ptolemy.
The diagram at the 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:
d
i
s
t
a
n
c
e
m
o
o
n
=
d
i
s
t
a
n
c
e
o
b
s
e
r
v
e
r
b
a
s
e
tan
⁡
(
a
n
g
l
e
)
{\displaystyle \mathrm {distance} _{\mathrm
{moon} }={\frac {\mathrm {distance} _{\mathrm
{observerbase} }}{\tan(\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 (with
the 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. 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. 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 Aristarchus' 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. This method was proposed
by Edmond Halley in 1716, although he did
not live to see the results. 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% above the currently accepted value,
149.6 million kilometers.
Much later, the Solar System was "scaled"
using the parallax of asteroids, some of which,
such as 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 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.
=== 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.
=== Dynamical parallax ===
Dynamical 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,
tan
⁡
p
=
1
AU
d
,
{\displaystyle \tan p={\frac {1{\text{ AU}}}{d}},}
where
p
{\displaystyle p}
is the parallax, 1 AU (149,600,000 km) is
approximately the average distance from the
Sun to Earth, and
d
{\displaystyle d}
is the distance to the star.
Using small-angle approximations (valid when
the angle is small compared 
to 1 radian),
tan
⁡
x
≈
x
radians
=
x
⋅
180
π
degrees
=
x
⋅
180
⋅
3600
π
arcseconds
,
{\displaystyle \tan x\approx x{\text{ radians}}=x\cdot
{\frac {180}{\pi }}{\text{ degrees}}=x\cdot
180\cdot {\frac {3600}{\pi }}{\text{ arcseconds}},}
so the parallax, measured in arcseconds, is
p
″
≈
1
AU
d
⋅
180
⋅
3600
π
.
{\displaystyle p''\approx {\frac {1{\text{
AU}}}{d}}\cdot 180\cdot {\frac {3600}{\pi
}}.}
If the parallax is 1", then the distance is
d
=
1
AU
⋅
180
⋅
3600
π
≈
206
,
265
AU
≈
3.2616
ly
≡
1
parsec
.
{\displaystyle d=1{\text{ AU}}\cdot 180\cdot
{\frac {3600}{\pi }}\approx 206,265{\text{
AU}}\approx 3.2616{\text{ ly}}\equiv 1{\text{
parsec}}.}
This defines the parsec, a convenient unit
for measuring distance using parallax. Therefore,
the distance, measured in parsecs, is simply
d
=
1
/
p
{\displaystyle d=1/p}
, when the parallax is given in arcseconds.
=== 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
δ
d
=
δ
(
1
p
)
=
|
∂
∂
p
(
1
p
)
|
δ
p
=
δ
p
p
2
{\displaystyle \delta d=\delta \left({1 \over
p}\right)=\left|{\partial \over \partial p}\left({1
\over p}\right)\right|\delta p={\delta p \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.
For meaningful results in stellar astronomy,
Dutch astronomer Floor van Leeuwen recommends
that the parallax error be no more than 10%
of the total parallax when computing this
error estimate.
=== Spatio-temporal parallax ===
From enhanced relativistic positioning systems,
spatio-temporal parallax generalizing the
usual notion of parallax in space only has
been developed. Then, eventfields in spacetime
can be deduced directly without intermediate
models of light bending by massive bodies
such as the one used in the PPN formalism
for instance.
== Metrology ==
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 own 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.
The newest iPhone includes an app that allows
measuring distances.
== Photogrammetry ==
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.
== 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). 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.
== Weapon sights ==
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. 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. 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 50m to 200m
without needing further adjustment.
=== Optical sights ===
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 that
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, 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 basically consists
of a movable optical element that enables
the optical system to shift the focus of the
target image at varying distances into exactly
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 which
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
100 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.
Scopes for guns with shorter practical ranges,
such as airguns, rimfire rifles, shotguns
and muzzleloaders, will have parallax settings
for shorter distances, commonly 50 yd/m for
rimfire scopes and 100 yd/m for shotguns and
muzzleloaders. 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 yards (2.7 metres).Non-magnifying
reflector or "reflex" sights have the ability
to 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 eye position in the cylindrical column
of light created by the collimating optics.
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. Some manufactures market reflector
sight models they call "parallax free," 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.
== Artillery gunfire ==
Because of the positioning of field or naval
artillery guns, 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 gun converges on the target.
== 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,
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 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.
== See also ==
Disparity
Lutz–Kelker bias
Parallax mapping, in computer graphics
Parallax scrolling, in computer graphics
Refraction, a visually similar principle caused
by water, etc.
Spectroscopic parallax
Triangulation, wherein a point is calculated
given its angles from other known points
Trigonometry
True range multilateration, wherein a point
is calculated given its distances from other
known points
Xallarap
== Notes
