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A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path ...
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes.Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles.
They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods that assume a spherical Earth, such as great-circle distance. The first (direct) method computes the location of a point that is a given distance and azimuth (direction) from another point. The second (inverse) method ...
Consider the great circle that contains the side BC. This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as A is (conventionally) termed the pole of A and it is denoted by A'.
The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. 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 R n + 1. Half of a great circle may be called a great ...
For a sphere the solutions to these problems are simple exercises in spherical trigonometry, whose solution is given by formulas for solving a spherical triangle. (See the article on great-circle navigation.) For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by Clairaut (1735).
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The shortest distance along the surface of a sphere between two points on the surface is along the great-circle which contains the two points. The great-circle distance article gives the formula for calculating the shortest arch length on a sphere about the size of the Earth. That article includes an example of the calculation.
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