# Great-circle navigation

**Great-circle navigation** or **orthodromic navigation** (related to **orthodromic course**; from the Greek *ορθóς*, right angle, and *δρóμος*, 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.^{[1]}

The great circle path may be found using spherical trigonometry; this is the spherical version of the *inverse geodesic problem*.
If a navigator begins at *P*_{1} = (φ_{1},λ_{1}) and plans to travel the great circle to a point at point *P*_{2} = (φ_{2},λ_{2}) (see Fig. 1, φ is the latitude, positive northward, and λ is the longitude, positive eastward), the initial and final courses α_{1} and α_{2} are given by formulas for solving a spherical triangle

where λ_{12} = λ_{2} − λ_{1}^{[note 1]}
and the quadrants of α_{1},α_{2} 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, σ_{12}, is given by

(The numerator of this formula contains the quantities that were used to determine
tanα_{1}.)
The distance along the great circle will then be *s*_{12} = *R*σ_{12}, where *R* is the assumed radius
of the earth and σ_{12} is expressed in radians.
Using the mean earth radius, *R* = *R*_{1} ≈ 6,371 km (3,959 mi) yields results for
the distance *s*_{12} which are within 1% of the
geodesic distance for the WGS84 ellipsoid.

To find the way-points, that is the positions of selected points on the great circle between
*P*_{1} and *P*_{2}, 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 λ_{0} — see Fig 1. The azimuth at this point, α_{0}, is given by

Let the angular distances along the great circle from *A* to *P*_{1} and *P*_{2} be σ_{01} and σ_{02} respectively. Then using Napier's rules we have

Finally, calculate the position and azimuth at an arbitrary point, *P* (see Fig. 2), by the spherical version of the *direct geodesic problem*.^{[note 5]} Napier's rules give

The atan2 function should be used to determine
σ_{01},
λ, and α.
For example, to find the
midpoint of the path, substitute σ = ^{1}⁄_{2}(σ_{01} + σ_{02}); alternatively
to find the point a distance *d* from the starting point, take σ = σ_{01} + *d*/*R*.
Likewise, the *vertex*, the point on the great
circle with greatest latitude, is found by substituting σ = +^{1}⁄_{2}π.
It may be convenient to parameterize the route in terms of the longitude using

These formulas apply to a spherical model of the earth. They are also used in solving for the great circle
on the *auxiliary sphere* which is a device for finding the shortest path, or *geodesic*, on
an ellipsoid of revolution; see
the article on geodesics on an ellipsoid.

Compute the great circle route from Valparaíso,
φ_{1} = −33°,
λ_{1} = −71.6°, to
Shanghai,
φ_{2} = 31.4°,
λ_{2} = 121.8°.

The formulas for course and distance give
λ_{12} = −166.6°,^{[note 8]}
α_{1} = −94.41°,
α_{2} = −78.42°, and
σ_{12} = 168.56°. Taking the earth radius to be
*R* = 6371 km, the distance is
*s*_{12} = 18743 km.

To compute points along the route, first find
α_{0} = −56.74°,
σ_{1} = −96.76°,
σ_{2} = 71.8°,
λ_{01} = 98.07°, and
λ_{0} = −169.67°.
Then to compute the midpoint of the route (for example), take
σ = ^{1}⁄_{2}(σ_{1} + σ_{2}) = −12.48°, and solve
for
φ = −6.81°,
λ = −159.18°, and
α = −57.36°.

If the geodesic is computed accurately on the WGS84 ellipsoid,^{[4]} the results
are α_{1} = −94.82°, α_{2} = −78.29°, and
*s*_{12} = 18752 km. The midpoint of the geodesic is
φ = −7.07°, λ = −159.31°,
α = −57.45°.

A straight line drawn on a gnomonic chart would be a great circle track. When this is transferred to a Mercator chart, it becomes a curve. The positions are transferred at a convenient interval of longitude and this is plotted on the Mercator chart.