# 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]}

## Course

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

^{[note 2]}^{[note 3]}

(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.

## Finding way-points

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

^{[note 4]}

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

- (If φ
_{1}= 0 and α_{1}= ^{1}⁄_{2}π, use σ_{01}= 0).

This gives σ_{01}, whence σ_{02} = σ_{01} + σ_{12}.

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*.^{[note 5]} Napier's rules give

^{[note 6]}

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

^{[note 7]}

Latitudes at regular intervals of longitude can be found and the resulting positions transferred to the Mercator chart allowing the great circle to be approximated by a series of rhumb lines. The path determined in this way gives the great ellipse joining the end points, provided the coordinates are interpreted as geographic coordinates on the ellipsoid.

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.

## Example

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°.

## Gnomonic chart

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.

## See also

- Compass rose
- Great circle
- Great-circle distance
- Great ellipse
- Geodesics on an ellipsoid
- Geographical distance
- Isoazimuthal
- Loxodromic navigation
- Map
- Marine sandglass
- Rhumb line
- Spherical trigonometry
- Windrose network

## Notes

**^**In the article on great-circle distances, the notation Δλ = λ_{12}and Δσ = σ_{12}is used. The notation in this article is needed to deal with differences between other points, e.g., λ_{01}.**^**A simpler formula is_{12}small.**^**These equations for α_{1},α_{2},σ_{12}are suitable for implementation on modern calculators and computers. For hand computations with logarithms, Delambre's analogies^{[2]}were usually used:^{[3]}refers to these equations as being in "logarithmic form", by which he means that all the trigonometric terms appear as products; this minimizes the number of table lookups required. Furthermore, the redundancy in these formulas serves as a check in hand calculations. If using these equations to determine the shorter path on the great circle, it is necessary to ensure that |λ_{12}| ≤ π (otherwise the longer path is found).**^**A simpler formula is_{0}≈ ±^{1}⁄_{2}π.**^**The direct geodesic problem, finding the position of*P*_{2}given*P*_{1}, α_{1}, and*s*_{12}, can also be solved by formulas for solving a spherical triangle, as follows,**^**A simpler formula is^{1}⁄_{2}π**^**The following is used:**^**λ_{12}is reduced to the range [−180°, 180°] by adding or subtracting 360° as necessary

## References

**^**Adam Weintrit; Tomasz Neumann (7 June 2011).*Methods and Algorithms in Navigation: Marine Navigation and Safety of Sea Transportation*. CRC Press. pp. 139–. ISBN 978-0-415-69114-7.**^**Todhunter, I. (1871).*Spherical Trigonometry*(3rd ed.). MacMillan. p. 26.**^**McCaw, G. T. (1932). "Long lines on the Earth".*Empire Survey Review*.**1**(6): 259–263. doi:10.1179/sre.1932.1.6.259.**^**Karney, C. F. F. (2013). "Algorithms for geodesics".*J. Geodesy*.**87**(1): 43–55. doi:10.1007/s00190-012-0578-z.

## External links

- Great Circle – from MathWorld Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
- Great Circle Mapper Interactive tool for plotting great circle routes.
- Great Circle Calculator deriving (initial) course and distance between two points.
- Great Circle Distance Graphical tool for drawing great circles over maps. Also shows distance and azimuth in a table.
- Google assistance program for orthodromic navigation