Great-circle navigation

Great-circle navigation or orthodromic navigation (related to orthodromic course; ) 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 tanα1.) 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 ≈ 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. To find the way-points, that is the positions of selected points on the great circle between P1 and P2, we first extrapolate the great circle back to its node A, the point at which the great circle crosses the equator in the northward direction: let the longitude of this point be λ0 — see Fig 1. The azimuth at this point, α0, is given by Let the angular distances along the great circle from A to P1 and P2 be σ01 and σ02 respectively. Then using Napier's rules we have (If φ1 = 0 and α1 = π, use σ01 = 0). This gives σ01, whence σ02 = σ01 + σ12. The longitude at the node is found from Finally, calculate the position and azimuth at an arbitrary point, P (see Fig. 2), by the spherical version of the direct geodesic problem.