Misplaced Pages

Revision as of 07:14, 1 June 2016

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. The word horizon derives from the Greek "ὁρίζων κύκλος" horizōn kyklos, "separating circle", from the verb ὁρίζω horizō, "to divide", "to separate", and that from "ὅρος" (oros), "boundary, landmark".

Appearance and usage

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 (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

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 longer distances 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.

However, ignoring the effect of atmospheric refraction, distance to the horizon from an observer close to the Earth's surface is about

${\displaystyle d\approx 3.57{\sqrt {h}}\,,}$

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

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)Cite error: A <ref> tag is missing the closing </ref> (see the help page).

${\displaystyle d={{R}_{\text{E}}}\left(\psi +\delta \right)\,,}$

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

${\displaystyle \cos \psi ={\frac {{R}_{\text{E}}{\mu }_{0}}{\left({{R}_{\text{E}}}+h\right)\mu }}\,,}$

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

${\displaystyle \delta =-\int _{0}^{h}{\tan \phi {\frac {{\text{d}}\mu }{\mu }}}\,,}$

where ${\displaystyle \phi \,\!}$ is the angle between the ray and a line through the center of the Earth. The angles ψ and ${\displaystyle \phi \,\!}$ are related by

${\displaystyle \phi =90{}^{\circ }-\psi \,.}$

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 R_E. The distance to the horizon is then

${\displaystyle d={\sqrt {2R^{\prime }h}}\,.}$

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

${\displaystyle d\approx 3.86{\sqrt {h}}\,;}$

with d in mi and h in ft,

${\displaystyle d\approx 1.32{\sqrt {h}}\,.}$

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

This section has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2013) (Learn how and when to remove this message) This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (June 2013) (Learn how and when to remove this message) This section needs expansion with: examples and additional citations. You can help by adding to it. (June 2013) (Learn how and when to remove this message)

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 ${\displaystyle \kappa }$, the altitude and the Earth's radius. It is

${\displaystyle \kappa ={\sqrt {\left(1+{\frac {h}{R}}\right)^{2}-1}}\ .}$

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 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 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. The Horizon curves by: sqrt(radius^2 + distance^2)-radius, equivalent to distance^2/R*2. At 100 km, it descends 784m.

Vanishing points

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...

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 2007 book Geometry of an Art, 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.

See also

• Aerial landscape art
• Atmospheric refraction
• Dawn
• Dusk
• Horizontal and vertical
• Landscape
  • Landscape art
• Limb
• Sextant

References

1. "offing", Webster's Third New International Dictionary, Unabridged. Pronounced, "Hor-I-zon".
2. "ὁρίζων", Henry George Liddell and Robert Scott, A Greek-English Lexicon. On Perseus Digital Library. Accessed 19 April 2011.
3. "ὁρίζω", Liddell and Scott, A Greek-English Lexicon.
4. "ὅρος", Liddell and Scott, A Greek-English Lexicon.
5. ^ Young, Andrew T. "Distance to the Horizon". Green Flash website (Sections: Astronomical Refraction, Horizon Grouping). San Diego State University Department of Astronomy. Retrieved April 16, 2011. {{cite news}}: Italic or bold markup not allowed in: |work= (help)
6. Brook Taylor (1719) New Principles of Perspective, p. v, as found in Kirsti Andersen (1991) Brook Taylor’s Work on Linear Perspective, p. 151, Springer, ISBN 0-387-97486-5
7. John Stillwell (2006) Yearning for the Impossible, Horizon, pp 47 to 76, A K Peters, Ltd., ISBN 1-56881-254-X

Further reading

• 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. {{cite news}}: Italic or bold markup not allowed in: |work= (help)

External links

• Horizontal coordinate system
• Celestial coordinate system