Great circle

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Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere. It is the largest circle that can be drawn on a given sphere.

Great circles serve as the analogue of "straight lines" in spherical geometry

Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
. See also spherical trigonometry

Spherical trigonometry

Spherical trigonometry is a part of spherical geometry that deals with polygons on the sphere and explains how to find relations between the involved angles....
and geodesic

Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
.

The great circle, also known as the Riemannian circle

Riemannian circle

In metric space theory and Riemannian geometry, the term Riemannian circle refers to a great circle equipped with its great-circle distance. In more detail, the term refers to the circle equipped with its intrinsic Riemannian metric of a compact 1-dimensional manifold of total length 2π, as opposed to the extrinsic metric obtaine...
, is the path with the smallest curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
, and hence, an arc (or an orthodrome) of a great circle is the shortest path between two points on the surface.

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Encyclopedia

Spherical geometry

Spherical trigonometry

Spherical trigonometry is a part of spherical geometry that deals with polygons on the sphere and explains how to find relations between the involved angles....
and geodesic

Geodesic

Riemannian circle

Curvature

Great-circle distance

The great-circle distance is the shortest distance between any two Point s on the surface of a sphere measured along a path on the surface of the sphere ....
. The great-circle route is the shortest path between two points on a sphere; however, if one were to travel along such a route, it would be difficult to steer manually as the heading would constantly be changing (except in the case of due north, south, or along the equator). Thus, Great Circle routes are often broken into a series of shorter Rhumb line

Rhumb line

In navigation, a rhumb line is a line crossing all meridian at the same angle, i.e. a path of constant bearing . Unlike a great circle route , following a rhumb line requires turning the vehicle more and more sharply while approaching the poles....
s which allow the use of constant headings between waypoint

Waypoint

A waypoint is a reference point in physical space used for purposes of navigation....
s along the Great Circle.

When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection

Mercator projection

The Mercator projection is a Map projection#Triangular presented by the Flemish people geographer and cartographer Gerardus Mercator, in 1569....
), they often look curved. This is because they lie on great circles. A route that would look like a straight line on the map would actually be longer. An exception is the gnomonic projection

Gnomonic projection

The gnomonic map projection displays all great circles as straight lines.Thus the shortest route between two locations in reality corresponds to that on the map....
, in which all straight lines represent great circles.

On the Earth, the meridians

Meridian (geography)

A meridian is an imaginary arc on the Earth's surface from the North Pole to the South Pole that connects all locations running along it with a given longitude....
are on great circles, and the equator

Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
is a great circle. Other lines of latitude

Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
are not great circles, because they are smaller than the equator; their centers are not at the center of the Earth -- they are small circle

Small circle

A small circle of a sphere is the circle constructed by a plane crossing the sphere not in its center. Small circles always have smaller diameters than the sphere itself ....
s instead. Great circles on Earth are roughly 40,000 km in length, though the Earth is not a perfect sphere; for instance, the equator is 40,075 km.

Some examples of great circles on the celestial sphere

Celestial sphere

In astronomy and navigation, the celestial sphere is an imagination rotation sphere of "gigantic radius", concentric spheres and coaxial with the Earth....
include the horizon

Horizon

The horizon is the apparent line that separates earth from sky.More precisely, it is the line that divides all of the directions one can possibly look into two categories: those which intersect the Earth's surface, and those which do not....
(in the astronomical sense), the celestial equator

Celestial equator

The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space....
, and the ecliptic

Ecliptic

The ecliptic is the apparent path that the Sun traces out in the sky during the year. As it appears to move in the sky in relation to the stars, the apparent path aligns with the planets throughout the course of the year....
.

Great circle routes are used by ships and aircraft where currents and winds are not a significant factor. For aircraft traveling west between continents in the northern hemisphere these paths will extend northward near or into the Arctic

Arctic

The Arctic is the region around the Earth's North Pole, opposite the Antarctica region around the South Pole. The Arctic includes the Arctic Ocean and parts of Canada, Greenland , Russia, the United States , Iceland, Norway, Sweden and Finland....
region, while easterly flights will often fly a more southerly track to take advantage of the jet stream

Jet stream

Jet streams are fast flowing, narrow thermal winds found at the tropopause, the transition between the troposphere and the stratosphere ,and are located at 10-15 kilometers above the surface of the Earth....
.

External links

Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
Interactive tool for plotting great circle routes.
Draws Great Circle routes between airports using the NASA Blue Marble as the base map.
deriving (initial) course and distance between two points.
Graphical tool for drawing great circles over maps. Also shows distance and azimuth in a table.
by John Snyder with additional contributions by Jeff Bryant, Pratik Desai, and Carl Woll, Wolfram Demonstrations Project
Wolfram Demonstrations Project

The Wolfram Demonstrations Project is a website developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience....
.