Great circle


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

A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center. (A small circle is the intersection of the sphere and a plane which does not pass through the center.) Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, and have the same center as the sphere. A great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean space is a great circle of exactly one sphere.

For any two points on the surface of a sphere there is a great circle through the two points. 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 spherical 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, namely the great-circle distance. The 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 two-planes that pass through the origin in the Euclidean space Rn+1.

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Earth geodesics

Strictly speaking the Earth is not a perfect sphere (it is an oblate spheroid or ellipsoid – i.e., slightly compressed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle. The equatorial radius of the earth is about 6378.137 kilometers. The polar radius of the earth is about 6356.752 kilometers (about 21.4 kilometers less). Nevertheless, the sphere model can be considered a first approximation.

When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection), they often look curved. This is because they lie on great circles. A route that would appear as a straight line on the map would actually be longer. An exception is the gnomonic projection, in which all straight lines represent great circles.

On a spherical Earth, the meridians (or lines of longitude) are great circles, as is the equator. Lines of latitude are not great circles, because they are smaller than the equator; their centers are not at the center of the Earth -- they are small circles instead. Since Earth is not a perfect sphere, the equator (which is generally considered a spherical great circle) is about 40,075 km, while a north-south meridian line (which is an ellipse) is almost 40,008 km. The quadratic mean or root mean square of these extremes provides a decent approximation of the average great-circle circumference, about 40041.5 km.[1]

Great-circle route from China to Canada

Some examples of great circles on the celestial sphere include the celestial horizon, the celestial equator, and the ecliptic.

Great circle routes are used by ships and aircraft where currents and winds are not a significant factor. Flight lengths can therefore often be approximated to the great-circle distance between two airports. For aircraft travelling west between continents in the northern hemisphere these paths will extend northward near or into the Arctic region, however easterly flights will often fly a more southerly track to take advantage of the jet stream.

For navigational convenience, Great Circle routes are often broken into a series of shorter rhumb lines which allow the use of constant headings between waypoints along the Great Circle.

Derivation of shortest paths

Airline routes between San Francisco and Tokyo following the most direct great circle (top), but following the jet stream (bottom) when heading eastwards

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

Consider the class of all regular paths from a point p to another point q. Introduce spherical coordinates so that p 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

\theta = \theta(t),\quad \phi = \phi(t),\quad a\le t\le b

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


ds=r\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.

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


S[\gamma]=r\int_a^b\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.

Note that S[γ] does not exceed the length of the meridian from p to q:

S[\gamma] \ge r\int_a^b|\theta'(t)|\,dt \ge r|\theta(q)-\theta(p)|.

Since the starting point and ending point are fixed, S is minimized if and only if φ' = 0, so the curve must lie on a meridian of the sphere φ = φ0 = constant. In Cartesian coordinates, this is

xsin ϕ0ycos ϕ0 = 0

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

See also

References

External links


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