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Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
''ορθóς'', right angle, and ''δρóμος'', path) is the practice of navigating a vessel (a
ship A ship is a large watercraft that travels the world's oceans and other sufficiently deep waterways, carrying cargo or passengers, or in support of specialized missions, such as defense, research, and fishing. Ships are generally distinguishe ...
or
aircraft An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines. ...
) along a
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry ...
. Such routes yield the shortest
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between two points on the globe.

# Course

The great circle path may be found using
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
; this is the spherical version of the '' inverse geodetic problem''. If a navigator begins at ''P''1 = (φ11) and plans to travel the great circle to a point at point ''P''2 = (φ22) (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 :$\begin \tan\alpha_1&=\frac,\\ \tan\alpha_2&=\frac,\\ \end$ where λ12 = λ2 − λ1In the article on
great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
s, the notation Δλ = λ12 and Δσ = σ12 is used. The notation in this article is needed to deal with differences between other points, e.g., λ01.
and the quadrants of α12 are determined by the signs of the numerator and denominator in the tangent formulas (e.g., using the
atan2 In computing and mathematics, the function atan2 is the 2- argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive
function). The
central angle A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc l ...
between the two points, σ12, is given by :$\tan\sigma_=\frac.$ (The numerator of this formula contains the quantities that were used to determine tanα1.) The distance along the great circle will then be ''s''12 = ''R''σ12, where ''R'' is the assumed radius of the earth and σ12 is expressed in
radians The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
. Using the mean earth radius, ''R'' = ''R''1 ≈  yields results for the distance ''s''12 which are within 1% of the geodesic length for the
WGS84 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
ellipsoid; see Geodesics on an ellipsoid for details.

# 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 In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex). Node may refer to: In mathematics *Vertex (graph theory), a vertex in a mathematical graph *Vertex (geometry), a point where two or more curves, lines, ...
'' ''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 An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematica ...
at this point, α0, is given by :$\tan\alpha_0 = \frac .$ Let the angular distances along the great circle from ''A'' to ''P''1 and ''P''2 be σ01 and σ02 respectively. Then using Napier's rules we have :$\tan\sigma_ = \frac \qquad$(If φ1 = 0 and α1 = π, use σ01 = 0). This gives σ01, whence σ02 = σ01 + σ12. The longitude at the node is found from :$\begin \tan\lambda_ &= \frac,\\ \lambda_0 &= \lambda_1 - \lambda_. \end$ Finally, calculate the position and azimuth at an arbitrary point, ''P'' (see Fig. 2), by the spherical version of the ''direct geodesic problem''. Napier's rules give :$\tan\phi = \frac ,$ :$\begin \tan\left(\lambda - \lambda_0\right) &= \frac ,\\ \tan\alpha &= \frac . \end$ The
atan2 In computing and mathematics, the function atan2 is the 2- argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive
function should be used to determine σ01, λ, and α. For example, to find the midpoint of the path, substitute σ = (σ01 + σ02); alternatively to find the point a distance ''d'' from the starting point, take σ = σ01 + ''d''/''R''. Likewise, the ''vertex'', the point on the great circle with greatest latitude, is found by substituting σ = +π. It may be convenient to parameterize the route in terms of the longitude using :$\tan\phi = \cot\alpha_0\sin\left(\lambda-\lambda_0\right).$ 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 line In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north. Introduction The effect of following a rhumb l ...
s. The path determined in this way gives the great ellipse joining the end points, provided the coordinates $\left(\phi,\lambda\right)$ 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 Valparaíso (; ) is a major city, seaport, naval base, and educational centre in the commune of Valparaíso, Chile. "Greater Valparaíso" is the second largest metropolitan area in the country. Valparaíso is located about northwest of Santiago ...
, φ1 = −33°, λ1 = −71.6°, to
Shanghai Shanghai (; , , Standard Mandarin pronunciation: ) is one of the four direct-administered municipalities of the People's Republic of China (PRC). The city is located on the southern estuary of the Yangtze River, with the Huangpu River flowi ...
, φ2 = 31.4°, λ2 = 121.8°. The formulas for course and distance give λ12 = −166.6°,λ12 is reduced to the range minus;180°, 180°by adding or subtracting 360° as necessary α1 = −94.41°, α2 = −78.42°, and σ12 = 168.56°. Taking the
earth radius Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, den ...
to be ''R'' = 6371 km, the distance is ''s''12 = 18743 km. To compute points along the route, first find α0 = −56.74°, σ01 = −96.76°, σ02 = 71.8°, λ01 = 98.07°, and λ0 = −169.67°. Then to compute the midpoint of the route (for example), take σ = (σ01 + σ02) = −12.48°, and solve for φ = −6.81°, λ = −159.18°, and α = −57.36°. If the geodesic is computed accurately on the
WGS84 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descr ...
ellipsoid, the results are α1 = −94.82°, α2 = −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 Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek l ...
and this is plotted on the Mercator chart.

*
Compass rose A compass rose, sometimes called a wind rose, rose of the winds or compass star, is a figure on a compass, map, nautical chart, or monument used to display the orientation of the cardinal directions (north, east, south, and west) and their in ...
*
Great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry ...
*
Great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
* Great ellipse * Geodesics on an ellipsoid *
Geographical distance Geographical distance or geodetic distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitud ...
*
Isoazimuthal The isoazimuth is the locus of the points on the Earth's surface whose initial orthodromic course with respect to a fixed point is constant. That is, if the initial orthodromic course Z from the starting point ''S'' to the fixed point ''X'' is 80 ...
Map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Altho ...
** Portolan map *
Marine sandglass A marine sandglass is a timepiece of simple design that is a relative of the common hourglass, a marine (nautical) instrument known since the 14th century (although reasonably presumed to be of very ancient use and origin). Sandglasses were used ...
*
Rhumb line In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north. Introduction The effect of following a rhumb l ...
*
Spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
* Windrose network

# References

{{reflist

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.

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.