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
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; this is the spherical version of the ''
inverse geodetic 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[In the article on great-circle distances,
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 α
1,α
2 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 le ...
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 ≈ 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 desc ...
ellipsoid; see
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an '' oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...
for details.
Finding way-points
To find the
way-point A waypoint is an intermediate point or place on a route or line of travel, a stopping point or point at which course is changed, the first use of the term tracing to 1880. In modern terms, it most often refers to coordinates which specify one's posi ...
s, 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 λ
0 — 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.
Mathematicall ...
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
:
(If φ
1 = 0 and α
1 = 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''. Napier's rules give
:
:
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
σ
01,
λ, and α.
For example, to find the
midpoint of the path, substitute σ = (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 = +π.
It may be convenient to parameterize the route in terms of the longitude using
:
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
150px, A spheroid
A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve ...
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
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an '' oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...
.
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 ...
,
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°,
[λ12
is reduced to the range minus;180°, 180°by adding or subtracting 360° as
necessary]
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
01 = −96.76°,
σ
02 = 71.8°,
λ
01 = 98.07°, and
λ
0 = −169.67°.
Then to compute the midpoint of the route (for example), take
σ = (σ
01 + σ
02) = −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 desc ...
ellipsoid,
[
] 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
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 thei ...
*
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
*
Great ellipse
150px, A spheroid
A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve ...
*
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an '' oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...
*
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 longitude. ...
*
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 8 ...
*
Loxodromic navigation
Loxodromic navigation (from Greek ''λοξóς'', oblique, and ''δρóμος'', path) is a method of navigation by following a rhumb line, a curve on the surface of the Earth that follows the same angle at the intersection with each meridian. This ...
*
Map
**
Portolan map
Portolan charts are nautical charts, first made in the 13th century in the Mediterranean basin and later expanded to include other regions. The word ''portolan'' comes from the Italian ''portulano'', meaning "related to ports or harbors", and w ...
*
Marine sandglass
*
Rhumb line
*
Spherical trigonometry
*
Windrose network
A rhumbline network (or windrose network) is a navigational aid consisting in multiple lines in a web-like grid drawn on portolan charts (and other early maps). These nautical charts were used in the medieval age and age of exploration in marin ...
Notes
References
{{reflist
External links
Great Circle – from MathWorldGreat 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 DistanceGraphical tool for drawing great circles over maps. Also shows distance and azimuth in a table.
Google assistance program for orthodromic navigation
Navigation
Circles
Spherical curves