Great-circle navigation or orthodromic navigation (related to orthodromic course; from Ancient Greekορθός (orthós) 'right angle' and δρόμος (drómos) '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
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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}^{[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
tan α_{1}.)
The distance along the great circle will then be s_{12} = Rσ_{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 length for the WGS84 ellipsoid; see Geodesics on an ellipsoid for details.
Relation to geocentric coordinate system
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Detailed evaluation of the optimum direction is possible if the sea surface is approximated by a sphere surface. The standard computation places the ship at a geodetic latitudeφ_{s} and geodetic longitudeλ_{s}, where φ is considered positive if north of the equator, and where λ is considered positive if east of Greenwich. In the geocentric coordinate system centered at the center of the sphere, the Cartesian components are
The minimum distanced is the distance along a great circle that runs through s and t. It is calculated in a plane that contains the sphere center and the great circle,
$d_{s,t}=R\theta _{s,t}$
where θ is the angular distance of two points viewed from the center of the sphere, measured in radians. The cosine of the angle is calculated by the dot product of the two vectors
If the ship steers straight to the North Pole, the travel distance is
$d_{s,N}=R\theta _{s,N}=R(\pi /2-\varphi _{s})$
If a ship starts at t and swims straight to the North Pole, the travel distance is
$d_{t,N}=R\theta _{t,n}=R(\pi /2-\varphi _{t})$
Derivation
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The cosine formula of spherical trigonometry^{[4]} yields for the
angle p between the great circles through s that point to the North on one hand and to t on the other hand
Because the brief derivation gives an angle between 0 and π which does not reveal the sign (west or east of north ?), a more explicit derivation is desirable which yields separately the sine and the cosine of p such that use of the correct branch of the inverse tangent allows to produce an angle in the full range -π≤p≤π.
The computation starts from a construction of the great circle between s and t. It lies in the plane that contains the sphere center, s and t and is constructed rotating s by the angle θ_{s,t} around an axis ω. The axis is perpendicular to the plane of the great circle and computed by the normalized vector cross product of the two positions:
The angle p is given by splitting this direction along two orthogonal directions in the plane tangential to the sphere at the point s. The two directions are given by the partial derivatives of s with respect to φ and with respect to λ, normalized to unit length:
where the positive sign means the positive position angles are defined to be north over east. The values of the cosine and sine of p are computed by multiplying this equation on both sides with the two unit vectors,
Instead of inserting the convoluted expression of s_{⊥}, the evaluation may employ that the triple product is invariant under a circular shift
of the arguments:
If atan2 is used to compute the value, one can reduce both expressions by division through cos φ_{t}
and multiplication by sin θ_{s,t},
because these values are always positive and that operation does not change signs; then effectively
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 nodeA, 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
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 = +1⁄2π.
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 joining the end points, provided the coordinates $(\phi ,\lambda )$
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
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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_{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 = 1⁄2(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 ellipsoid,^{[5]} 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
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A straight line drawn on a gnomonic chart is a portion of a great circle. When this is transferred to a Mercator chart, it becomes a curve. The positions are transferred at a convenient interval of longitude and this track is plotted on the Mercator chart for navigation.
^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}.
however, this is numerically less accurate if σ_{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:
McCaw^{[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).
^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,
The solution for way-points given in the main text is more general than this solution in that
it allows
way-points at specified longitudes to be found. In addition, the solution for σ
(i.e., the position of the node)
is needed when finding geodesics on an ellipsoid by means of the auxiliary sphere.
^
The following is used: $\cos \sigma =\cos \phi \cos(\lambda -\lambda _{0})$
^λ_{12}
is reduced to the range [−180°, 180°] by adding or subtracting 360° as
necessary
References
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^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". Journal of Geodesy. 87 (1): 43–55. arXiv:1109.4448. Bibcode:2013JGeod..87...43K. doi:10.1007/s00190-012-0578-z.
External links
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Great Circle – from MathWorld Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
Great Circle Map Interactive tool for plotting great circle routes on a sphere.
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