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Great-circle navigation
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Great-circle navigation
Great-circle navigation or orthodromic navigation (related to orthodromic course; from Ancient Greek ορθός (orthós) 'right angle' and δρόμος (drómos) 'path') is the practice of navigating a vessel (a ship or aircraft) along a great circle. Such routes yield the shortest distance between two points on the globe.
The great circle path may be found using spherical trigonometry; this is the spherical version of the inverse geodetic problem. If a navigator begins at P1 = (φ1,λ1) and plans to travel the great circle to a point at point P2 = (φ2,λ2) (see Fig. 1, φ is the latitude, positive northward, and λ is the longitude, positive eastward), the initial and final courses α1 and α2 are given by formulas for solving a spherical triangle
where λ12 = λ2 − λ1 and the quadrants of α1,α2 are determined by the signs of the numerator and denominator in the tangent formulas (e.g., using the atan2 function). The central angle between the two points, σ12, is given by
(The numerator of this formula contains the quantities that were used to determine tan α1.) The distance along the great circle will then be s12 = Rσ12, where R is the assumed radius of the Earth and σ12 is expressed in radians. Using the mean Earth radius, R = R1 ≈ 6,371 km (3,959 mi) yields results for the distance s12 which are within 1% of the geodesic length for the WGS84 ellipsoid; see Geodesics on an ellipsoid for details.
Detailed evaluation of the optimum direction is possible if the sea surface is approximated by a sphere surface. The standard computation places the ship at a geodetic latitude φs and geodetic longitude λs, where φ is considered positive if north of the equator, and where λ is considered positive if east of Greenwich. In the geocentric coordinate system centered at the center of the sphere, the Cartesian components are
and the target position is
The North Pole is at
The minimum distance d is the distance along a great circle that runs through s and t. It is calculated in a plane that contains the sphere center and the great circle,
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Great-circle navigation AI simulator
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Great-circle navigation
Great-circle navigation or orthodromic navigation (related to orthodromic course; from Ancient Greek ορθός (orthós) 'right angle' and δρόμος (drómos) 'path') is the practice of navigating a vessel (a ship or aircraft) along a great circle. Such routes yield the shortest distance between two points on the globe.
The great circle path may be found using spherical trigonometry; this is the spherical version of the inverse geodetic problem. If a navigator begins at P1 = (φ1,λ1) and plans to travel the great circle to a point at point P2 = (φ2,λ2) (see Fig. 1, φ is the latitude, positive northward, and λ is the longitude, positive eastward), the initial and final courses α1 and α2 are given by formulas for solving a spherical triangle
where λ12 = λ2 − λ1 and the quadrants of α1,α2 are determined by the signs of the numerator and denominator in the tangent formulas (e.g., using the atan2 function). The central angle between the two points, σ12, is given by
(The numerator of this formula contains the quantities that were used to determine tan α1.) The distance along the great circle will then be s12 = Rσ12, where R is the assumed radius of the Earth and σ12 is expressed in radians. Using the mean Earth radius, R = R1 ≈ 6,371 km (3,959 mi) yields results for the distance s12 which are within 1% of the geodesic length for the WGS84 ellipsoid; see Geodesics on an ellipsoid for details.
Detailed evaluation of the optimum direction is possible if the sea surface is approximated by a sphere surface. The standard computation places the ship at a geodetic latitude φs and geodetic longitude λs, where φ is considered positive if north of the equator, and where λ is considered positive if east of Greenwich. In the geocentric coordinate system centered at the center of the sphere, the Cartesian components are
and the target position is
The North Pole is at
The minimum distance d is the distance along a great circle that runs through s and t. It is calculated in a plane that contains the sphere center and the great circle,
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