Horizon

Saturday, 27 October 2012

Horizon

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Burj Khalifa
Burj Khalifa ( "Khalifa Tower"), known as Burj Dubai prior to its inauguration, is a skyscraper in Dubai, United Arab Emirates, and the tallest man-made structure ever built, at . Construction began on 21 September 2004, with the exterior of the structure completed on 1 October 2009. The building officially opened on 4 January 2010, and is part of the new flagship development called Downtown Dubai at the 'First Interchange' along Sheikh Zayed Road, near Dubai's main business district.
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Great South Bay
Great South Bay is a lagoon that's situated between Long Island and Fire Island, in the State of New York. It is approximately 45 miles (72 km) long. It's protected from the Atlantic Ocean by Fire Island, a barrier island, as well as the eastern end of Jones Beach Island and Captree Island.
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Long Island
Long Island is an island located in southeastern New York, United States, just east of Manhattan. Stretching northeast into the Atlantic Ocean, Long Island contains four counties, two of which are boroughs (Queens and Brooklyn) of New York City, and two of which (Nassau and Suffolk) are mainly suburban or rural. In popular usage, the term "Long Island" generally refers only to Nassau and Suffolk counties in order to differentiate them from New York City, though all four counties on the island are part of the New York metropolitan area.
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Wisconsin
Wisconsin () is a U.S. state located in the north-central United States and is considered part of the Midwest. It is bordered by Minnesota to the west, Iowa to the southwest, Illinois to the south, Lake Michigan to the east, Upper Michigan to the northeast, and Lake Superior to the north. Wisconsin's capital is Madison, and its largest city is Milwaukee. As of 2009 the state has an estimated 5.6 million residents. The state contains 72 counties.
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Abraham Bosse
Abraham Bosse (c. 1602-1604 – February 14, 1676) was a French artist, mainly as a printmaker in etching, but also in watercolour.
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David Hilbert
David Hilbert (; January 23, 1862 –
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Girard Desargues
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John Stillwell
John Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University. He is the author of many textbooks and other books on mathematics including:
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Pappus of Alexandria
Pappus of Alexandria (Greek ) (c. 290 – c. 350) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection (c. 340), and for Pappus's Theorem in projective geometry. Nothing is known of his life, except (from his own writings) that he had a son named Hermodorus, and was a teacher in Alexandria.
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26 Oct 2012

Related topics virgin auctionvirginity auction Advertisement A 20 year-old Brazilian student has sold her virginity for the sum of $780,000 through an online auction on a website simply called Virgins...

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Kim Jong-il Funeral ceremony .Kim Jong-il died of a suspected heart attack on 17 December 2011 at 08:30 while traveling by train to an area outside Pyongyang.

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Novosti

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MOSCOW, October 25 (RIA Novosti) - A high-ranking North Korean Army official was reportedly executed with a mortar round for drinking liquor during the 100-day...

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File - President Barack Obama disembarks Air Force One upon arrival at Joplin Regional Airport in Joplin, Missouri, May 21, 2012.

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Article by WN.com Correspondent Dallas Darling By the time President Richard Nixon told his Chinese hosts during a banquet toast, "Let us start a long march together, not in lockstep, but on different...

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Waves, brought by Hurricane Sandy, crash on a house in the Caribbean Terrace neighborhood in eastern Kingston, Jamaica, Wednesday, Oct. 24, 2012.

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Myanmar Buddhist monks march, protesting against plans to open office of the Organization of the Islamic Conference (OIC), in Yangon, Myanmar, Monday, Oct. 15, 2012.

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Stephanie Gilmore's life one big holiday 27 Oct 2012

The Daily Telegraph Australia FIVE-times world surfing champ Stephanie Gilmore shamelessly admits her life is one big holiday. Gilmore, who lives in Tweed Heads, in northern NSW, is plotting a way to tack some time on Italy's Amalfi coast on to her next competition in France. She loves to take photographs of all the places she...(size: 3.9Kb)

Family fun on the Gold Coast 27 Oct 2012

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Form, not rotation: how to restore nation's Test glory 27 Oct 2012

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As one school door closes, another opens 27 Oct 2012

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INSIGHT-BP oil spill settlement before election seen unlikely 27 Oct 2012

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US presidential candidates cool on warming 27 Oct 2012

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For other uses, see Horizon (disambiguation).

A water horizon, in northern Wisconsin, U.S.

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]

1 Appearance and usage
2 Distance to the horizon
3 Curvature of the horizon
4 Projective geometry
5 See also
6 Notes and References
7 External links

Appearance and usage [link]

View of Earth's horizon as seen from Space Shuttle Endeavour, 2002

Historically, the distance to the visible horizon at sea has been extremely important as it represented the maximum range of communication and vision before the development of the radio and the telegraph. 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 [link]

Ignoring the effect of atmospheric refraction, distance to the horizon from an observer close to the Earth's surface is about^[5]

Failed to parse (Missing texvc executable; please see math/README to configure.): d \approx 3.57\sqrt{h} \,,

where d is in kilometres and h is height above sea level in metres.

Examples:

For an observer standing on the ground with h = 1.70 metres (5 ft 7 in) (average eye-level height), 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 39 kilometres (24 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 111 kilometres (69 mi).

With d in miles^[6] and h in feet,

Failed to parse (Missing texvc executable; please see math/README to configure.): d \approx 1.22\sqrt{h} \,.

Examples, assuming no refraction:

For an observer on the ground with eye level at h = 5 ft 7 in (5.583 ft), 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).

Geometrical model [link]

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 sphere with no atmosphere then the distance to the horizon can easily be calculated. (The earth's radius of curvature actually varies by 1%, so this formula isn't exact even assuming no refraction.)

The secant tangent theorem states that

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): d^2 = h(D+h)\,\!

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): (R+h)^2 = R^2 + d^2 \,\!

Failed to parse (Missing texvc executable; please see math/README to configure.): R^2 + 2Rh + h^2 = R^2 + d^2 \,\!

Failed to parse (Missing texvc executable; please see math/README to configure.): 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,

Failed to parse (Missing texvc executable; please see math/README to configure.): s = R \gamma \,;

then

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos \gamma = \cos\frac{s}{R}=\frac{R}{R+h}\,.

Solving for s gives

Failed to parse (Missing texvc executable; please see math/README to configure.): 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,

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan \gamma = \frac {d} {R} \,;

substituting for γ and rearranging gives

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 [link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): d = \sqrt{2Rh} \,.

Using metric units and taking the radius of the Earth as 6371 km, the distance to the horizon is

Failed to parse (Missing texvc executable; please see math/README to configure.): d \approx \sqrt{12.74h} \approx 3.57\sqrt{h} \,,

where d is in kilometres, and h is the height of the eye of the observer above ground or sea level in metres.

Using imperial units, the distance to the horizon is

Failed to parse (Missing texvc executable; please see math/README to configure.): d \approx \sqrt{1.50h} \approx 1.22 \sqrt{h} \,,

where d is in miles and h is in feet.

These formulas may be used when h is much smaller than the radius of the Earth (6371 km), including all views from any mountaintops, aeroplanes, 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 [link]

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 [link]

Geometrical horizon distance

To compute the height of an object visible 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 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 lighthouse will be visible from the boat if

Failed to parse (Missing texvc executable; please see math/README to configure.): D_\mathrm{BL} < 3.57\,(\sqrt{h_\mathrm{B}} + \sqrt{h_\mathrm{L}}) \,,

where D_BL is in kilometres and h_B and h_L are in metres. If atmospheric refraction is considered, the visibility condition becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): D_\mathrm{BL} < 3.86\,(\sqrt{h_\mathrm{B}} + \sqrt{h_\mathrm{L}}) \,.

Effect of atmospheric refraction [link]

Because of atmospheric refraction of light rays, the actual distance to the horizon is slightly greater than the distance calculated with geometrical formulas. With standard atmospheric conditions, the difference is about 8%; however, refraction is strongly affected by temperature gradients, which can vary considerably from day to day, especially over water, so calculated values for refraction are only approximate.^[5]

Rigorous method—Sweer
The distance d to the horizon is given by^[7]

Failed to parse (Missing texvc executable; please see math/README to configure.): d={{R}_{\text{E}}}\left( \psi +\delta \right) \,,

where R_E 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta =-\int_{0}^{h}{\tan \phi \frac{\text{d}\mu }{\mu }} \,,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,\!

is the angle between the ray and a line through the center of the Earth. The angles ψ and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,\!
are related by

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi =90{}^\circ -\psi \,.

Simple method—Young
A much simpler approach uses the geometrical model but uses a radius R′ = 7/6 R_E. The distance to the horizon is then^[5]

Failed to parse (Missing texvc executable; please see math/README to configure.): d=\sqrt{2 R^\prime h} \,.

Taking the radius of the Earth as 6371 km, with d in km and h in m,

Failed to parse (Missing texvc executable; please see math/README to configure.): d \approx 3.86 \sqrt{h} \,;

with d in mi and h in ft,

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 [link]

A man dives against the horizon into the Great South Bay of Long Island

From a point above the surface the horizon appears slightly bent (it is a circle, after all). There is a basic geometrical relationship between this visual curvature Failed to parse (Missing texvc executable; please see math/README to configure.): \kappa , the altitude and the Earth's radius. It is

Failed to parse (Missing texvc executable; please see math/README to configure.): \kappa=\sqrt{\left(\frac{R+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 45° 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. 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.

Projective geometry [link]

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.

In visual geometry, the horizon is a provocative concept. Standing on a floor plane where parallel lines converge toward a point on the horizon, one sees that the point of convergence on the horizon is a vanishing point, which geometers call a point at infinity. Since each point on the horizon corresponds to a convergence point for its own set of parallel lines, the horizon is a line at infinity that represents the various sets of parallel lines. The science of graphical perspective developed in conjunction with formal projective geometry. John Stillwell describes these developments in the chapter titled "Horizon" in his book Yearning for the Impossible (2006). After introducing parallelism through traditional axioms, he introduces coordinates, which are "a natural consequence of the parallel axiom", and the slope of a line. Then moving to visual perspective of a a tiled floor, he reviews the construzione legittima in 1505 by Jean Pèlerin . Projective configurations provide a context to discuss "incidence" as a geometric primitive. There is a short review of literature by Girard Desargues, Etienne Pascal, Abraham Bosse, and Phillipe de la Hire. Projective configurations associated with Desargues and Pappus of Alexandria are illustrated. The role of these configurations as both theorems and axioms is discussed. Stillwell also ventures 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, is considered along with supporting work by David Hilbert (1899) and Ruth Moufang (1932). Concluding the chapter with four axioms for the projective plane that determine Euclidean geometry as well as the laws of algebra, Stillwell writes

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.^[8]

Notes and References [link]

^ "offing", Webster's Third New International Dictionary, Unabridged.
^ "ὁρίζων", Henry George Liddell and Robert Scott, A Greek-English Lexicon. On Perseus Digital Library. Accessed 19 April 2011.
^ "ὁρίζω", Liddell and Scott, A Greek-English Lexicon.
^ "ὅρος", Liddell and Scott, A Greek-English Lexicon.
^ ^a ^b ^c Andrew T. Young, "Distance to the Horizon". Accessed 16 April 2011.
^ In this article, mile refers to a "land" mile of 5,280 feet (1,609.344 m).
^ John Sweer, "The Path of a Ray of Light Tangent to the Surface of the Earth", Journal of the Optical Society of America, 28 (September 1938):327–29. Available as paid download.
^ John Stillwell (2006) Yearning for the Impossible, pp 47 to 76, A K Peters, Ltd., ISBN 1-56881-254-X

External links [link]

Derivation of the distance to the horizon. Steve Sque.
Dip of the Horizon. Andrew T. Young.

This page contains text from Wikipedia, the Free Encyclopedia - http://en.wikipedia.org/wiki/Horizon

This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.

Branches of Mathematics

Pure mathematics

Applied mathematics

Related Topics

Event Horizon (1997)

"Event Horizon (1997)" Actors

"Event Horizon (1997)" Director

Horizon Filmography

Make changes yourself !

Contents

Appearance and usage [link]

Distance to the horizon [link]

Geometrical model [link]

Approximate geometrical formulas [link]

Exact formula for a spherical Earth [link]

Objects above the horizon [link]

Effect of atmospheric refraction [link]

Curvature of the horizon [link]

Projective geometry [link]

See also [link]

Notes and References [link]

External links [link]

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http://fr.wn.com/Horizon (physique)

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