Horizon
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For other uses, see Horizon (disambiguation).
A High desert horizon at sunset, California, USA
The horizon or skyline is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting intersection of earth and sky is called the visible horizon. When looking at a sea from a shore, the part of the sea closest to the horizon is called the offing.[1] The word horizon derives from the Greek "ὁρίζων κύκλος" horizōn kyklos, "separating circle",[2] from the verb ὁρίζω horizō, "to divide", "to separate",[3] and that from "ὅρος" (oros), "boundary, landmark".[4]
Contents
[hide]
1 Appearance and usage
2 Distance to the horizon
2.1 Other planets
2.2 Geometrical model
2.3 Approximate geometrical formulas
2.4 Exact formula for a spherical Earth
2.5 Objects above the horizon
2.6 Effect of atmospheric refraction
3 Curvature of the horizon
4 Vanishing points
5 See also
6 References
7 Further reading
8 External links
Appearance and usage[edit]
View of the ocean with a ship near to the horizon
Historically, the distance to the visible horizon has long been vital to survival and successful navigation, especially at sea, because it determined an observer's maximum range of vision and thus of communication, with all the obvious consequences for safety and the transmission of information that this range implied. This importance lessened with the development of the radio and the telegraph, but even today, when flying an aircraft under visual flight rules, a technique called attitude flying is used to control the aircraft, where the pilot uses the visual relationship between the aircraft's nose and the horizon to control the aircraft. A pilot can also retain his or her spatial orientation by referring to the horizon.
In many contexts, especially perspective drawing, the curvature of the Earth is disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the observer increases. For observers near sea level the difference between this geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is imperceptible to the naked eye[dubious – discuss] (but for someone on a 1000-meter hill looking out to sea the true horizon will be about a degree below a horizontal line).
In astronomy the horizon is the horizontal plane through the eyes of the observer. It is the fundamental plane of the horizontal coordinate system, the locus of points that have an altitude of zero degrees. While similar in ways to the geometrical horizon, in this context a horizon may be considered to be a plane in space, rather than a line on a picture plane.
Distance to the horizon[edit]
One typically sees further along the Earth's curved surface than a simple geometric calculation allows for because of refraction error. If the ground, or water, surface is colder than the air above it, a cold, dense layer of air forms close to the surface, causing light to be refracted downward as it travels, and therefore, to some extent, to go around the curvature of the Earth. The reverse happens if the ground is hotter than the air above it, as often happens in deserts, producing mirages. As an approximate compensation for refraction, surveyors measuring distances longer than 300 feet subtract 14% from the calculated curvature error and ensure lines of sight are at least 5 feet from the ground, to reduce random errors created by refraction.
Typical desert horizon
However, ignoring the effect of atmospheric refraction, distance to the horizon from an observer close to the Earth's surface is about[5]
d ≈ 3.57 h , {\displaystyle d\approx 3.57{\sqrt {h}}\,,}
where d is in kilometres and h is height above ground level in metres. The constant 3.57 has units of km/m½.
Examples:
For an observer standing on the ground with h = 1.70 metres (5 ft 7 in), the horizon is at a distance of 4.7 kilometres (2.9 mi).
For an observer standing on the ground with h = 2 metres (6 ft 7 in), the horizon is at a distance of 5 kilometres (3.1 mi).
For an observer standing on a hill or tower of 100 metres (330 ft) in height, the horizon is at a distance of 36 kilometres (22 mi).
For an observer standing at the top of the Burj Khalifa (828 metres (2,717 ft) in height), the horizon is at a distance of 103 kilometres (64 mi).
For an observer atop Mount Everest (8,848 metres (29,029 ft) in altitude), the horizon is at a distance of 336 kilometres (209 mi).
With d in miles (i.e. "land miles" of 5,280 feet (1,609.344 m)[5]) and h in feet,
d ≈ 1.5 h ≈ 1.22 h . {\displaystyle d\approx {\sqrt {1.5h}}\approx 1.22{\sqrt {h}}\,.}
Where the constant 1.22 has units of mi/ft½. Examples, assuming no refraction:
For an observer on the ground with eye level at h = 5 ft 7 in (1.70 m), the horizon is at a distance of 2.9 miles (4.7 km).
For an observer standing on a hill or tower 100 feet (30 m) in height, the horizon is at a distance of 12.2 miles (19.6 km).
For an observer on the summit of Aconcagua (22,841 feet (6,962 m) in height), the sea-level horizon to the west is at a distance of 184 miles (296 km).
For a U-2 pilot, whilst flying at its service ceiling 70,000 feet (21,000 m), the horizon is at a distance of 324 miles (521 km)
Other planets[edit]
On terrestrial planets and other solid celestial bodies with negligible atmospheric effects, the distance to the horizon for a "standard observer" varies as the square root of the planet's radius. Thus, the horizon on Mercury is 62% as far away from the observer as it is on Earth, on Mars the figure is 73%, on the Moon the figure is 52%, on Mimas the figure is 18%, and so on.
Geometrical model[edit]
Geometrical basis for calculating the distance to the horizon, secant tangent theorem
Geometrical distance to the horizon, Pythagorean theorem
Three types of horizon
If the Earth is assumed to be a featureless sphere (rather than an oblate spheroid) with no atmospheric refraction, then the distance to the horizon can easily be calculated.[6]
The secant-tangent theorem states that
O C 2 = O A × O B . {\displaystyle \mathrm {OC} ^{2}=\mathrm {OA} \times \mathrm {OB} \,.}
Make the following substitutions:
d = OC = distance to the horizon
D = AB = diameter of the Earth
h = OB = height of the observer above sea level
D+h = OA = diameter of the Earth plus height of the observer above sea level
The formula now becomes
d 2 = h ( D + h ) {\displaystyle d^{2}=h(D+h)\,\!}
or
d = h ( D + h ) = h ( 2 R + h ) , {\displaystyle d={\sqrt {h(D+h)}}={\sqrt {h(2R+h)}}\,,}
where R is the radius of the Earth.
The equation can also be derived using the Pythagorean theorem. Since the line of sight is a tangent to the Earth, it is perpendicular to the radius at the horizon. This sets up a right triangle, with the sum of the radius and the height as the hypotenuse. With
d = distance to the horizon
h = height of the observer above sea level
R = radius of the Earth
referring to the second figure at the right leads to the following:
( R + h ) 2 = R 2 + d 2 {\displaystyle (R+h)^{2}=R^{2}+d^{2}\,\!}
R 2 + 2 R h + h 2 = R 2 + d 2 {\displaystyle R^{2}+2Rh+h^{2}=R^{2}+d^{2}\,\!}
d = h ( 2 R + h ) . {\displaystyle d={\sqrt {h(2R+h)}}\,.}
Another relationship involves the distance s along the curved surface of the Earth to the horizon; with γ in radians,
s = R γ ; {\displaystyle s=R\gamma \,;}
then
cos ⁡ γ = cos ⁡ s R = R R + h . {\displaystyle \cos \gamma =\cos {\frac {s}{R}}={\frac {R}{R+h}}\,.}
Solving for s gives
s = R cos − 1 ⁡ R R + h . {\displaystyle s=R\cos ^{-1}{\frac {R}{R+h}}\,.}
The distance s can also be expressed in terms of the line-of-sight distance d; from the second figure at the right,
tan ⁡ γ = d R ; {\displaystyle \tan \gamma ={\frac {d}{R}}\,;}
substituting for γ and rearranging gives
s = R tan − 1 ⁡ d R . {\displaystyle s=R\tan ^{-1}{\frac {d}{R}}\,.}
The distances d and s are nearly the same when the height of the object is negligible compared to the radius (that is, h ≪ R).
Approximate geometrical formulas[edit]
Graphs of distances to the true horizon on Earth for a given height h. s is along the surface of the Earth, d is the straight line distance, and ~d is the approximate straight line distance assuming h << the radius of the Earth, 6371 km. In the SVG image, hover over a graph to highlight it.
If the observer is close to the surface of the earth, then it is valid to disregard h in the term (2R + h), and the formula becomes-
d = 2 R h . {\displaystyle d={\sqrt {2Rh}}\,.}
Using kilometres for d and R, and metres for h, and taking the radius of the Earth as 6371 km, the distance to the horizon is
d ≈ 2 ⋅ 6371 ⋅ h / 1000 ≈ 3.570 h , {\displaystyle d\approx {\sqrt {2\cdot 6371\cdot {h/1000}}}\approx 3.570{\sqrt {h}}\,,} .
Using imperial units, with d and R in statute miles (as commonly used on land), and h in feet, the distance to the horizon is
d ≈ 2 ⋅ 3963 ⋅ h / 5280 ≈ 1.5 h ≈ 1.22 h {\displaystyle d\approx {\sqrt {2\cdot 3963\cdot {h/5280}}}\approx {\sqrt {1.5h}}\approx 1.22{\sqrt {h}}} .
If d is in nautical miles, and h in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving:
d ≈ h {\displaystyle d\approx {\sqrt {h}}}
These formulas may be used when h is much smaller than the radius of the Earth (6371 km or 3959 mi), including all views from any mountaintops, airplanes, or high-altitude balloons. With the constants as given, both the metric and imperial formulas are precise to within 1% (see the next section for how to obtain greater precision).
Exact formula for a spherical Earth[edit]
If h is significant with respect to R, as with most satellites, then the approximation made previously is no longer valid, and the exact formula is required:
d = 2 R h + h 2 , {\displaystyle d={\sqrt {2Rh+h^{2}}}\,,}
where R is the radius of the Earth (R and h must be in the same units). For example, if a satellite is at a height of 2000 km, the distance to the horizon is 5,430 kilometres (3,370 mi); neglecting the second term in parentheses would give a distance of 5,048 kilometres (3,137 mi), a 7% error.
Objects above the horizon[edit]
Geometrical horizon distance
To compute the greatest distance at which an observer can see the top of an object above the horizon, compute the distance to the horizon for a hypothetical observer on top of that object, and add it to the real observer's distance to the horizon. For example, for an observer with a height of 1.70 m standing on the ground, the horizon is 4.65 km away. For a tower with a height of 100 m, the horizon distance is 35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than 40.35 km away. Conversely, if an observer on a boat (h = 1.7 m) can just see the tops of trees on a nearby shore (h = 10 m), the trees are probably about 16 km away.
Referring to the figure at the right, the top of the lighthouse will be visible to a lookout in a crow's nest at the top of a mast of the boat if
D B L < 3.57 ( h B + h L ) , {\displaystyle D_{\mathrm {BL} }<3.57\,({\sqrt {h_{\mathrm {B} }}}+{\sqrt {h_{\mathrm {L} }}})\,,}
where DBL is in kilometres and hB and hL are in metres.
A view across a 20-km-wide bay in the coast of Spain. Note the curvature of the Earth hiding the base of the buildings on the far shore.
As another example, suppose an observer, whose eyes are two metres above the level ground, uses binoculars to look at a distant building which he knows to consist of thirty storeys, each 3.5 metres high. He counts the storeys he can see, and finds there are only ten. So twenty storeys or 70 metres of the building are hidden from him by the curvature of the Earth. From this, he can calculate his distance from the building:
D ≈ 3.57 ( 2 + 70 ) {\displaystyle D\approx 3.57({\sqrt {2}}+{\sqrt {70}})}
which comes to about 35 kilometres.
It is similarly possible to calculate how much of a distant object is visible above the horizon. Suppose an observer's eye is 10 metres above sea level, and he is watching a ship that is 20 km away. His horizon is:
3.57 10 {\displaystyle 3.57{\sqrt {10}}}
kilometres from him, which comes to about 11.3 kilometres away. The ship is a further 8.7 km away. The height of a point on the ship that is just visible to the observer is given by:
h ≈ ( 8.7 3.57 ) 2 {\displaystyle h\approx \left({\frac {8.7}{3.57}}\right)^{2}}
which comes to almost exactly six metres. The observer can therefore see that part of the ship that is more than six metres above the level of the water. The part of the ship that is below this height is hidden from him by the curvature of the Earth. In this situation, the ship is said to be hull-down.
Effect of atmospheric refraction[edit]
If the Earth were an airless world like the Moon, the above calculations would be accurate. However, Earth has an atmosphere of air, whose density and refractive index vary considerably depending on the temperature and pressure. This makes the air refract light to varying extents, affecting the appearance of the horizon. Usually, the density of the air just above the surface of the Earth is greater than its density at greater altitudes. This makes its refractive index greater near the surface than at higher altitudes, which causes light that is travelling roughly horizontally to be refracted downward.[7] This makes the actual distance to the horizon greater than the distance calculated with geometrical formulas. With standard atmospheric conditions, the difference is about 8%. This changes the factor of 3.57, in the metric formulas used above, to about 3.86. This correction can be, and often is, applied as a fairly good approximation when conditions are close to standard. When conditions are unusual, this approximation fails. Refraction is strongly affected by temperature gradients, which can vary considerably from day to day, especially over water. In extreme cases, usually in springtime, when warm air overlies cold water, refraction can allow light to follow the Earth's surface for hundreds of kilometres. Opposite conditions occur, for example, in deserts, where the surface is very hot, so hot, low-density air is below cooler air. This causes light to be refracted upward, causing mirage effects that make the concept of the horizon somewhat meaningless. Calculated values for the effects of refraction under unusual conditions are therefore only approximate.[5] Nevertheless, attempts have been made to calculate them more accurately than the simple approximation described above.
Outside the visual wavelength range, refraction will be different. For radar (e.g. for wavelengths 300 to 3 mm i.e. frequencies between 1 and 100 GHz) the radius of the Earth may be multiplied by 4/3 to obtain an effective radius giving a factor of 4.12 in the metric formula i.e. the radar horizon will be 15% beyond the geometrical horizon or 7% beyond the visual. The 4/3 factor is not exact, as in the visual case the refraction depends on atmospheric conditions.
Integration method—Sweer
If the density profile of the atmosphere is known, the distance d to the horizon is given by[8]
d = R E ( ψ + δ ) , {\displaystyle d={{R}_{\text{E}}}\left(\psi +\delta \right)\,,}
where RE is the radius of the Earth, ψ is the dip of the horizon and δ is the refraction of the horizon. The dip is determined fairly simply from
cos ⁡ ψ = R E μ 0 ( R E + h ) μ , {\displaystyle \cos \psi ={\frac {{R}_{\text{E}}{\mu }_{0}}{\left({{R}_{\text{E}}}+h\right)\mu }}\,,}
where h is the observer's height above the Earth, μ is the index of refraction of air at the observer's height, and μ0 is the index of refraction of air at Earth's surface.
The refraction must be found by integration of
δ = − ∫ 0 h tan ⁡ ϕ d μ μ , {\displaystyle \delta =-\int _{0}^{h}{\tan \phi {\frac {{\text{d}}\mu }{\mu }}}\,,}
where ϕ {\displaystyle \phi \,\!} is the angle between the ray and a line through the center of the Earth. The angles ψ and ϕ {\displaystyle \phi \,\!} are related by
ϕ = 90 ∘ − ψ . {\displaystyle \phi =90{}^{\circ }-\psi \,.}
Simple method—Young
A much simpler approach, which produces essentially the same results as the first-order approximation described above, uses the geometrical model but uses a radius R′ = 7/6 RE. The distance to the horizon is then[5]
d = 2 R ′ h . {\displaystyle d={\sqrt {2R^{\prime }h}}\,.}
Taking the radius of the Earth as 6371 km, with d in km and h in m,
d ≈ 3.86 h ; {\displaystyle d\approx 3.86{\sqrt {h}}\,;}
with d in mi and h in ft,
d ≈ 1.32 h . {\displaystyle d\approx 1.32{\sqrt {h}}\,.}
Results from Young's method are quite close to those from Sweer's method, and are sufficiently accurate for many purposes.
Curvature of the horizon[edit]
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The curvature of the horizon is easily seen in this photograph, taken from a space shuttle at an altitude of 226 km in 2008.
From a point above the surface the horizon appears slightly bent (it is a circle). There is a basic geometrical relationship between this visual curvature κ {\displaystyle \kappa } , the altitude h {\displaystyle h} , and the Earth's radius R {\displaystyle R} , . It is
κ = ( 1 + h R ) 2 − 1 . {\displaystyle \kappa ={\sqrt {\left(1+{\frac {h}{R}}\right)^{2}-1}}\ .}
The curvature is the reciprocal of the curvature angular radius in radians. A curvature of 1 appears as a circle of an angular radius of 57.3° corresponding to an altitude of approximately 2640 km above the Earth's surface. At an altitude of 10 km (33,000 ft, the typical cruising altitude of an airliner) the mathematical curvature of the horizon is about 0.056, the same curvature of the rim of circle with a radius of 10 m that is viewed from 56 cm directly above the center of the circle. However, the apparent curvature is less than that due to refraction of light in the atmosphere and because the horizon is often masked by high cloud layers that reduce the altitude above the visual surface.
Vanishing points[edit]
Two points on the horizon are at the intersections of the lines extending the segments representing the edges of the building in the foreground. The horizon line coincides here with the line at the top of the doors and windows.
Main article: Vanishing point
The horizon is a key feature of the picture plane in the science of graphical perspective. Assuming the picture plane stands vertical to ground, and P is the perpendicular projection of the eye point O on the picture plane, the horizon is defined as the horizontal line through P. The point P is the vanishing point of lines perpendicular to the picture. If S is another point on the horizon, then it is the vanishing point for all lines parallel to OS. But Brook Taylor (1719) indicated that the horizon plane determined by O and the horizon was like any other plane:
The term of Horizontal Line, for instance, is apt to confine the Notions of a Learner to the Plane of the Horizon, and to make him imagine, that that Plane enjoys some particular Privileges, which make the Figures in it more easy and more convenient to be described, by the means of that Horizontal Line, than the Figures in any other plane;…But in this Book I make no difference between the Plane of the Horizon, and any other Plane whatsoever...[9][10]
The peculiar geometry of perspective where parallel lines converge in the distance, stimulated the development of projective geometry which posits a point at infinity where parallel lines meet. In her book Geometry of an Art (2007), Kirsti Andersen described the evolution of perspective drawing and science up to 1800, noting that vanishing points need not be on the horizon. In a chapter titled "Horizon", John Stillwell recounted how projective geometry has led to incidence geometry, the modern abstract study of line intersection. Stillwell also ventured into foundations of mathematics in a section titled "What are the Laws of Algebra ?" The "algebra of points", originally given by Karl von Staudt deriving the axioms of a field was deconstructed in the twentieth century, yielding a wide variety of mathematical possibilities. Stillwell states
This discovery from 100 years ago seems capable of turning mathematics upside down, though it has not yet been fully absorbed by the mathematical community. Not only does it defy the trend of turning geometry into algebra, it suggests that both geometry and algebra have a simpler foundation than previously thought.[11]
See also[edit]
Aerial landscape art
Atmospheric refraction
Dawn
Dusk
Horizontal and vertical
Landscape
Landscape art
Limb[disambiguation needed]
Limb darkening
Lunar limb
Sextant
References[edit]
Jump up ^ "Offing". Webster's Third New International Dictionary (Unabridged ed.). Pronounced, "Hor-I-zon".
Jump up ^ Liddell, Henry George & Scott, Robert. "ὁρίζων". A Greek-English Lexicon. Perseus Digital Library. Archived from the original on June 5, 2011. Retrieved April 19, 2011. CS1 maint: Uses authors parameter (link)
Jump up ^ Liddell, Henry George & Scott, Robert. "ὁρίζω". A Greek-English Lexicon. Perseus Digital Library. Archived from the original on June 5, 2011. Retrieved April 19, 2011. CS1 maint: Uses authors parameter (link)
Jump up ^ Liddell, Henry George & Scott, Robert. "ὅρος". A Greek-English Lexicon. Perseus Digital Library. Archived from the original on June 5, 2011. Retrieved April 19, 2011. CS1 maint: Uses authors parameter (link)
^ Jump up to: a b c d Young, Andrew T. "Distance to the Horizon". Green Flash website (Sections: Astronomical Refraction, Horizon Grouping). San Diego State University Department of Astronomy. Archived from the original on October 18, 2003. Retrieved April 16, 2011.
Jump up ^ Plait, Phil (15 January 2009). "How far away is the horizon?". Discover. Bad Astronomy. Kalmbach Publishing Co. Archived from the original on 29 March 2017. Retrieved 2017-03-28.
Jump up ^ Proctor, Richard Anthony; Ranyard, Arthur Cowper (1892). Old and New Astronomy. Longmans, Green and Company. p. 73. Archived from the original on 2017-03-29.
Jump up ^ Sweer, John (1938). "The Path of a Ray of Light Tangent to the Surface of the Earth". Journal of the Optical Society of America. 28: 327–329. doi:10.1364/JOSA.28.000327. (Subscription required (help)).
Jump up ^ Taylor, Brook. New Principles of Perspective. p. 1719. CS1 maint: Uses authors parameter (link)
Jump up ^ Anderson, Kirsti (1991). "Brook Taylor's Work on Linear Perspective". Springer. p. 151. ISBN 0-387-97486-5.
Jump up ^ Stillwell, John (2006). "Yearning for the Impossible". Horizon. A K Peters, Ltd. pp. 47–76. ISBN 1-56881-254-X.
Further reading[edit]
Young, Andrew T. "Dip of the Horizon". Green Flash website (Sections: Astronomical Refraction, Horizon Grouping). San Diego State University Department of Astronomy. Retrieved April 16, 2011.
External links[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Horizon&oldid=817091142"
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mwe-math-element
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i
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mn
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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mw-headline Exact_formula_for_a_spherical_Earth
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mw-editsection-bracket
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/w/index.php?title=Horizon&action=edit&section=6 Edit section: Exact formula for a spherical Earth
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src class aria-hidden style alt
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mw-headline Objects_above_the_horizon
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/w/index.php?title=Horizon&action=edit&section=7 Edit section: Objects above the horizon
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/wiki/File:Horizon,_Valencia_(Spain).JPG image
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class
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a
href title
/wiki/Spain Spain
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local
a
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/wiki/Curvature_of_the_Earth mw-redirect Curvature of the Earth
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MJX-TeXAtom-ORD
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mw-headline Effect_of_atmospheric_refraction
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a
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/w/index.php?title=Horizon&action=edit&section=8 Edit section: Effect of atmospheric refraction
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/wiki/Refractive_index Refractive index
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#cite_note-ProctorRanyard1892-7
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/wiki/Mirage Mirage
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#cite_note-ATYoungDistToHoriz-5
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cite_ref-ATYoungDistToHoriz_5-2 reference
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/wiki/Radar Radar
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dl
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i
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#cite_note-Sweer1938-8
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cite_ref-Sweer1938_8-0 reference
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MJX-TeXAtom-ORD
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mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f1e6bf9cff21a0f728be6de3861ddd7368fd20 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:16.551ex; height:2.843ex; d={{R}_{\text{E}}}\left( \psi +\delta  \right) \,,
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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http://www.w3.org/1998/Math/MathML
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mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
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src class aria-hidden style alt
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local
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class
mwe-math-element
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dd
html
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sub
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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msubsup
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mtext
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class
MJX-TeXAtom-ORD
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html
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mfrac
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MJX-TeXAtom-ORD
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MJX-TeXAtom-ORD
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thinmathspace
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application/x-tex
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xmlns
http://www.w3.org/1998/Math/MathML
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mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
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src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9954796e0dd28ee663a667ba96f9f153b6a9ea mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:20.191ex; height:6.343ex; \delta =-\int_{0}^{h}{\tan \phi \frac{\text{d}\mu }{\mu }} \,,
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class
mwe-math-element
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application/x-tex
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xmlns
http://www.w3.org/1998/Math/MathML
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class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
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local
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src class aria-hidden style alt
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html
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class
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i
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mi
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class
MJX-TeXAtom-ORD
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application/x-tex
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local
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xmlns
http://www.w3.org/1998/Math/MathML
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local
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class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
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src class aria-hidden style alt
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html
local
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class
mwe-math-element
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local
p
html
local
mi
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local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
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mo
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class
MJX-TeXAtom-ORD
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width
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MJX-TeXAtom-ORD
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encoding
application/x-tex
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xmlns
http://www.w3.org/1998/Math/MathML
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class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
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local
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src class aria-hidden style alt
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html
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class
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dd
html
local
dl
html
local
dt
html
local
dl
html
local
i
html
local
i
html
local
sub
html
local
span
class
nowrap
html
local
a
href
#cite_note-ATYoungDistToHoriz-5
html
local
sup
id class
cite_ref-ATYoungDistToHoriz_5-3 reference
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mi
html
local
mi
class mathvariant
MJX-variant normal
html
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mrow
class
MJX-TeXAtom-ORD
html
local
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html
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mi
html
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msqrt
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class
MJX-TeXAtom-ORD
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width
thinmathspace
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mo
html
local
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displaystyle scriptlevel
true 0
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local
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class
MJX-TeXAtom-ORD
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local
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encoding
application/x-tex
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local
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html
local
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xmlns
http://www.w3.org/1998/Math/MathML
html
local
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class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/ef00209ce03f41f881cdf8477cd176d7985b8684 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:12.703ex; height:3.509ex; d=\sqrt{2 R^\prime h} \,.
html
local
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class
mwe-math-element
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local
dd
html
local
dl
html
local
i
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mi
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mspace
width
thinmathspace
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
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local
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class
MJX-TeXAtom-ORD
html
local
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encoding
application/x-tex
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local
semantics
html
local
math
xmlns
http://www.w3.org/1998/Math/MathML
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local
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class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/27c39a190dc1d46444676cc77bdeeb6b687e26ea mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:12.851ex; height:3.009ex; d \approx 3.86 \sqrt{h} \,;
html
local
span
class
mwe-math-element
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local
dd
html
local
dl
html
local
i
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mi
html
local
msqrt
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class
MJX-TeXAtom-ORD
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mspace
width
thinmathspace
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local
mo
html
local
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true 0
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class
MJX-TeXAtom-ORD
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local
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encoding
application/x-tex
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local
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local
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xmlns
http://www.w3.org/1998/Math/MathML
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class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
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src class aria-hidden style alt
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html
local
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class
mwe-math-element
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dd
html
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local
p
html
local
span
class id
mw-headline Curvature_of_the_horizon
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class
mw-editsection-bracket
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local
a
href title
/w/index.php?title=Horizon&action=edit&section=9 Edit section: Curvature of the horizon
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class
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style
width:52px
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class
mbox-image
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html
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mw-collapsible-text
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html
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class role tabindex
mw-collapsible-toggle button 0
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b
html
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a
class href
external text //en.wikipedia.org/w/index.php?title=Horizon&action=edit
html
local
b
html
local
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href title
/wiki/Talk:Horizon Talk:Horizon
html
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b
html
local
a
href title
/wiki/Help:Maintenance_template_removal Help:Maintenance template removal
html
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i
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small
html
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img
alt src width height srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png 50 39 //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x 512 399
html
local
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href class
/wiki/File:Question_book-new.svg image
html
local
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style
width:52px
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local
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class
mbox-image
html
local
a
href title
/wiki/Wikipedia:Citing_sources Wikipedia:Citing sources
html
local
a
href title
/wiki/Wikipedia:Verifiability Wikipedia:Verifiability
html
local
b
html
local
a
class href
external text //en.wikipedia.org/w/index.php?title=Horizon&action=edit
html
local
a
href title
/wiki/Help:Introduction_to_referencing_with_Wiki_Markup/1 Help:Introduction to referencing with Wiki Markup/1
html
local
a
href title
/wiki/Wikipedia:Verifiability#Burden_of_evidence Wikipedia:Verifiability
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class
hide-when-compact
html
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i
html
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small
html
local
a
href title
/wiki/Help:Maintenance_template_removal Help:Maintenance template removal
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i
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class
hide-when-compact
html
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class
mbox-text-span
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mbox-text
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tr
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tbody
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class role
plainlinks metadata ambox ambox-content ambox-Unreferenced presentation
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alt src width height srcset data-file-width data-file-height
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html
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href class
/wiki/File:Wiki_letter_w_cropped.svg image
html
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class
mbox-image
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b
html
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class href
external text //en.wikipedia.org/w/index.php?title=Horizon&action=edit&section=
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/wiki/Help:Maintenance_template_removal Help:Maintenance template removal
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class
hide-when-compact
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mbox-text-span
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plainlinks metadata ambox ambox-content ambox-multiple_issues compact-ambox presentation
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alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/3/37/A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg/220px-A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg 220 146 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/3/37/A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg/330px-A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg/440px-A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg 2x 3032 2008
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/wiki/File:A_colorful_view_of_airglow_layers_at_Earth%27s_horizon.jpg image
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