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.
Parallax is an apparent displacement or difference
of orientation 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. In
astronomy,
parallax is the only direct method by which distances to objects
(typically
stars) beyond the
Solar System can be measured. The
Hipparcos satellite has used the technique for
over 100,000 nearby stars. This provides the basis for all other
distance measurements in astronomy, the
cosmic distance ladder. Here, the
term "parallax" is the angle or semi-angle of inclination between
two sightlines 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 to use parallax to gain
depth perception; this process is
known as
stereopsis.
A simple everyday example of parallax can be seen in the dashboard
of motor vehicles that use a "needle" type speedometer gauge (when
the needle is mounted in front of its dial scale in a way that
leaves a noticeable spacing between them). When viewed from
directly in front, the speed may show 60 (i.e. the needle appears
against the '60' mark on the dial behind); but when viewed from the
passenger seat (i.e. from an oblique angle) the needle can appear
against a slightly lower or higher mark (depending on whether it is
viewed from the left or from the right), because of the combined
effect of the spacing and the angle of view.
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
0.77233 ± 0.00242 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
our galaxy.
The
European Space
Agency's Gaia mission, due to
launch in 2011 and come online in 2012, 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
Stellar parallax motion
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 (\mathrm{pc})
= 1 / p (\mathrm{arcsec}). For example, the distance to Proxima
Centauri is 1/0.772= .
Lunar parallax
Lunar parallax (often short for
lunar horizontal
parallax or
lunar equatorial horizontal parallax), is
a special case of parallax: the Moon seen from different viewing
positions on the Earth (at one given moment) can appear differently
placed against the background of fixed stars.
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 formerly, of
navigators), and the study of the way in which this coordinate
varies with time forms part of
lunar
theory.
Diagram of daily lunar parallax
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:
- \mathrm{distance}_{\textrm{moon}} = \frac
{\mathrm{distance}_{\mathrm{observerbase}}} {\tan
(\mathrm{angle})}
Example of lunar parallax: Occultation
of Pleiades by the Moon
This is the method referred to by
Jules
Verne in
From the
Earth to the Moon:
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).
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. 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 these 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.
Measuring Venus transit times to
determine solar parallax
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, like
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.
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,
- \sin p = \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),
- \sin x \approx x\textrm{\ radians} = x \cdot \frac {180} {\pi}
\textrm{\ degrees} = x \cdot 180 \cdot \frac {3600} {\pi} \textrm{\
arcseconds} ,
so the parallax, measured in arcseconds, is
- p \approx \frac {1 \textrm{\ AU}} {d} \cdot 180 \cdot
\frac{3600} {\pi} .
If the parallax is 1", then the distance is
- d = 1 \textrm{\ AU} \cdot 180 \cdot \frac {3600} {\pi} =
206,265 \textrm{\ AU} = 3.2616 \textrm{\ ly} \equiv 1 \textrm{\
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
- \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.
Visual perception
Because the eyes of humans and other highly evolved 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 animal (or just the
head) moves 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.
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 a
car's passenger than to the driver.
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
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.
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 (DS) still often use parallax.
In naval gunfire
Owing to the positioning of
gun turrets
on a
warship, each has a slightly different
perspective of the target relative to the ship itself. Therefore,
the ship's
system for aiming its
guns must compensate for parallax in order to assure that
fire from each turret converges on the
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."
The word is used in the title of
Alan
J. Pakula's
1974 movie The Parallax View, in which a
reporter (
Warren Beatty) investigates
an
assassination. The word in this
case refers to a fictional corporation portrayed in the film.
Notes
References
See also
External links