Great circle

A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

For most pairs of points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry. The length of the minor arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry where such great circles are called Riemannian circles. These great circles are the geodesics of the sphere.

In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn + 1.

Great circle hemispheres
A great circle divides the sphere in two equal hemispheres

Derivation of shortest paths

To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it.

Consider the class of all regular paths from a point to another point . Introduce spherical coordinates so that coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by

provided we allow to take on arbitrary real values. The infinitesimal arc length in these coordinates is

So the length of a curve from to is a functional of the curve given by

According to the Euler–Lagrange equation, is minimized if and only if

,

where is a -independent constant, and

From the first equation of these two, it can be obtained that

.

Integrating both sides and considering the boundary condition, the real solution of is zero. Thus, and can be any value between 0 and , indicating that the curve must lie on a meridian of the sphere. In Cartesian coordinates, this is

which is a plane through the origin, i.e., the center of the sphere.

Applications

Some examples of great circles on the celestial sphere include the celestial horizon, the celestial equator, and the ecliptic. Great circles are also used as rather accurate approximations of geodesics on the Earth's surface for air or sea navigation (although it is not a perfect sphere), as well as on spheroidal celestial bodies.

The equator of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the land and water hemispheres. A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point.

The Funk transform integrates a function along all great circles of the sphere.

See also

External links

125th meridian west

The meridian 125° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, North America, the Pacific Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 125th meridian west forms a great circle with the 55th meridian east.

130th meridian west

The meridian 130° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, North America, the Pacific Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 130th meridian west forms a great circle with the 50th meridian east.

155th meridian west

The meridian 155° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, North America, the Pacific Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 155th meridian west forms a great circle with the 25th meridian east.

165th meridian west

The meridian 165° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, North America, the Pacific Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 165th meridian west forms a great circle with the 15th meridian east.

175th meridian east

The meridian 175° east of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Asia, the Pacific Ocean, New Zealand, the Southern Ocean, and Antarctica to the South Pole.

The 175th meridian east forms a great circle with the 5th meridian west.

175th meridian west

The meridian 175° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Asia, the Pacific Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 175th meridian west forms a great circle with the 5th meridian east.

30th meridian west

The meridian 30° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Greenland, the Atlantic Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 30th meridian west forms a great circle with the 150th meridian east, and it is the reference meridian for the time zone UTC-2.

40th meridian west

The meridian 40° west of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Greenland, the Atlantic Ocean, South America, the Southern Ocean, and Antarctica to the South Pole.

The 40th meridian west forms a great circle with the 140th meridian east.

5th meridian east

The meridian 5° east of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Europe, Africa, the Atlantic Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 5th meridian east forms a great circle with the 175th meridian west.

65th meridian east

The meridian 65° east of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Europe, Asia, the Indian Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 65th meridian east forms a great circle with the 115th meridian west.

95th meridian east

The meridian 95° east of Greenwich is a line of longitude that extends from the North Pole across the Arctic Ocean, Asia, the Indian Ocean, the Southern Ocean, and Antarctica to the South Pole.

The 95th meridian east forms a great circle with the 85th meridian west.

Arc (geometry)

In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve. A common example in the plane (a two-dimensional manifold), is a segment of a circle called a circular arc. In space, if the arc is part of a great circle (or great ellipse), it is called a great arc.

Every pair of distinct points on a circle determines two arcs. If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than π radians (180 degrees), and the other arc, the major arc, will subtend an angle greater than π radians.

Great-circle distance

The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.

The determination of the great-circle distance is part of the more general problem of great-circle navigation, which also computes the azimuths at the end points and intermediate way-points.

Through any two points on a sphere that are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is called a Riemannian circle in Riemannian geometry.

Between two points that are directly opposite each other, called antipodal points, there are infinitely many great circles, and all great circle arcs between antipodal points have a length of half the circumference of the circle, or , where r is the radius of the sphere.

The Earth is nearly spherical (see Earth radius), so great-circle distance formulas give the distance between points on the surface of the Earth correct to within about 0.5%. (See Arc length § Arcs of great circles on the Earth.)

List of airports in Antarctica

This is a list of airports in Antarctica, sorted by location.

Mercator projection

The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical navigation because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects (making it a conformal map projection), the Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are, relative to landmasses near the equator such as Central Africa.

Meridian (geography)

A (geographical) meridian (or line of longitude) is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. The position of a point along the meridian is given by that longitude and its latitude, measured in angular degrees north or south of the Equator. Each meridian is perpendicular to all circles of latitude. Each is also the same length, being half of a great circle on the Earth's surface and therefore measuring 20,003.93 km (12,429.9 miles).

Newark Earthworks

The Newark Earthworks in Newark and Heath, Ohio, consist of three sections of preserved earthworks: the Great Circle Earthworks, the Octagon Earthworks, and the Wright Earthworks. This complex, built by the Hopewell culture between 100 CE and 500 CE, contains the largest earthen enclosures in the world, and was about 3,000 acres in total extent. Less than 10 percent of the total site has been preserved since European-American settlement; this area contains a total of 206 acres (83 ha). It is operated as a state park by the Ohio History Connection. A designated National Historic Landmark, in 2006 the Newark Earthworks was also designated as the "official prehistoric monument of the State of Ohio."This is part of the Hopewell Ceremonial Earthworks, one of 14 sites nominated in January 2008 by the U.S. Department of the Interior for potential submission by the US to the UNESCO World Heritage List.

Stanton Drew stone circles

The Stanton Drew stone circles are just outside the village of Stanton Drew in the English county of Somerset. The largest stone circle is the Great Circle, 113 metres (371 ft) in diameter and the second largest stone circle in Britain (after Avebury); it is considered to be one of the largest Neolithic monuments to have been built. The date of construction is not known but is thought to be between 3000 and 2000 BCE which places it in the Late Neolithic

to Early Bronze Age. It was made a scheduled monument in 1982.The Great Circle was surrounded by a ditch and is accompanied by smaller stone circles to the northeast and southwest. There is also a group of three stones, known as The Cove, in the garden of the local pub. Slightly further from the Great Circle is a single stone, known as Hautville's Quoit. Some of the stones are still vertical, but the majority are now recumbent and some are no longer present.

The stone circles have been studied since John Aubrey's visit in 1664, with some excavations of the site in the 18th century. In the late 20th and early 21st centuries geophysical surveys have confirmed the size of the stone circles and identified additional pits and postholes. The Cove has been shown to be around one thousand years older than the stone circles, so dating from 4000-3000 BCE. A variety of myths and legends about the stone circles have been recorded, including one about dancers at a celebration who have been turned to stone.

TransLink (South East Queensland)

TransLink is an agency of the Department of Transport and Main Roads first introduced by the Queensland Government in June 2003 to coordinate and integrate bus, ferry and rail services. TransLink works with Airtrain Citylink, Brisbane Transport, Transdev Brisbane Ferries, Queensland Rail and 17 other operators to provide these services in South East Queensland. TransLink operates an integrated ticketing system and the go card smartcard system to allow the use of one ticket on multiple services.

In July 2008 TransLink devolved from being a division of Queensland Transport to the more autonomous TransLink Transit Authority.

In November 2012 the Authority was dissolved and reabsorbed as a division of the Queensland Department of Transport and Main Roads. Combining the former Authority and qConnect, it is now responsible for the co-ordination of public transport across the whole of Queensland.

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