Great-circle navigation or orthodromic navigation (related to orthodromic course;) 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 geodetic problem.If a navigator begins at P1 = (φ1,λ1) and plans to travel the great circle to a point at point P2 = (φ2,λ2) (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{align} \tan\alpha | ||||
|
,\\ \tan\alpha | ||||
|
,\\ \end{align}
\tan\sigma12=
\sqrt{(\cos\phi1\sin\phi2-\sin\phi1\cos\phi2\cosλ12)2+(\cos\phi2\sinλ12)2 | |
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 latitude and geodetic longitude, where is considered positive if north of the equator, and where is considered positive if east of Greenwich. In the geocentric coordinate system centered at the center of the sphere, the Cartesian components are
{s}=R\left(\begin{array}{c}\cos\varphis\cosλs\\ \cos\varphis\sinλs\\ \sin\varphis \end{array}\right)
{t}=R\left(\begin{array}{c}\cos\varphit\cosλt\\ \cos\varphit\sinλt\\ \sin\varphit \end{array}\right).
{N}=R\left(\begin{array}{c}0\\ 0\\ 1 \end{array}\right).
ds,t=R\thetas,t
s ⋅ t=R2\cos\thetas,t=
2(\sin\varphi | |
R | |
s\sin\varphi |
t+\cos\varphis\cos\varphit\cos(λt-λs))
ds,N=R\thetas,N=R(\pi/2-\varphis)
dt,N=R\thetat,n=R(\pi/2-\varphit)
The cosine formula of spherical trigonometry yields for the angle between the great circles through that point to the North on one hand and to on the other hand
\cos\thetat,N=\cos\thetas,t\cos\thetas,N+\sin\thetas,t\sin\thetas,N\cosp.
\sin\varphit=\cos\thetas,t\sin\varphis+\sin\thetas,t\cos\varphis\cosp.
\sinp | |
\sin\thetat,N |
=
\sin(λt-λs) | |
\sin\thetas,t |
.
\sin\varphit=\cos\thetas,t\sin\varphis+
\sin(λt-λs) | |
\sinp |
\cos\varphit\cos\varphis\cosp;
\tanp=
\sin(λt-λs)\cos\varphit\cos\varphis | |
\sin\varphit-\cos\thetas,t\sin\varphis |
.
Because the brief derivation gives an angle between 0 and 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 such that use of the correct branch of the inverse tangent allows to produce an angle in the full range .
The computation starts from a construction of the great circle between and . It lies in the plane that contains the sphere center, and and is constructed rotating by the angle around an axis . The axis is perpendicular to the plane of the great circle and computed by the normalized vector cross product of the two positions:
\omega=
1 | |
R2\sin\thetas,t |
s x t=
1 | |
\sin\thetas,t |
\left(\begin{array}{c} \cos\varphis\sinλs\sin\varphit-\sin\varphis\cos\varphit\sinλt \\ \sin\varphis\cosλt\cos\varphit-\cos\varphis\sin\varphit\cosλs \\ \cos\varphis\cos\varphit\sin(λt-λs) \end{array}\right).
s\perp=\omega x
1 | |
R |
s=
1 | |
\sin\thetas,t |
\left(\begin{array}{c} \cos\varphit\cosλ
2λ | |
s)-\cosλ |
s(\sin\varphis\cos\varphis\sin\varphi
2\varphi | |
s\sinλ |
s\cos\varphit\sinλt)\\ \cos\varphit\sinλ
2λ | |
s)-\sinλ |
s(\sin\varphis\cos\varphis\sin\varphi
2\varphi | |
s\cosλ |
s\cos\varphit\cosλt)\\ \cos\varphis[\cos\varphis\sin\varphit-\sin\varphis\cos\varphit\cos(λt-λs)] \end{array}\right)
s(\theta)=\cos\thetas+\sin\thetas\perp, 0\le\theta\le2\pi.
\partial | |
\partial\theta |
s\mid=s\perp.
uN=\left(\begin{array}{c} -\sin\varphis\cosλs\\ -\sin\varphis\sinλs\\ \cos\varphis \end{array}\right);
uE=\left(\begin{array}{c} -\sinλs\\ \cosλs\\ 0 \end{array} \right);
uN ⋅ s=uE ⋅ uN=0
s\perp=\cospuN+\sinpuE
\cosp=s\perp ⋅
u | ||||
|
[\cos\varphis\sin\varphit-\sin\varphis\cos\varphit\cos(λt-λs)];
\sinp=s\perp ⋅
u | ||||
|
[\cos\varphit\sin(λt-λs)].
\cosp=(\omega x
1 | |
R |
s) ⋅ uN =\omega ⋅ (
1 | |
R |
s x uN).
If atan2 is used to compute the value, one can reduce both expressions by division through and multiplication by,because these values are always positive and that operation does not change signs; then effectively
\tanp=
\sin(λt-λs) | |
\cos\varphis\tan\varphit-\sin\varphis\cos(λt-λs) |
.
To find the way-points, that is the positions of selected points on the great circle betweenP1 and P2, we first extrapolate the great circle back to its node A, the pointat which the great circle crosses theequator in the northward direction: let the longitude of this point be λ0 — see Fig 1. The azimuth at this point, α0, is given by
\tan\alpha0=
\sin\alpha1\cos\phi1 | |||||||||||||||
|
\tan\sigma01=
\tan\phi1 | |
\cos\alpha1 |
This gives σ01, whence σ02 = σ01 + σ12.
The longitude at the node is found from
\begin{align} \tanλ01&=
\sin\alpha0\sin\sigma01 | |
\cos\sigma01 |
,\\ λ0&=λ1-λ01. \end{align}
{\color{white}. )}\tan\phi=
\cos\alpha0\sin\sigma | ||||||||||||
|
\begin{align} \tan(λ-λ0)&=
\sin\alpha0\sin\sigma | |
\cos\sigma |
,\\ \tan\alpha&=
\tan\alpha0 | |
\cos\sigma |
. \end{align}
\tan\phi=\cot\alpha0\sin(λ-λ0).
(\phi,λ)
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,φ1 = -33°,λ1 = -71.6°, toShanghai,φ2 = 31.4°,λ2 = 121.8°.
The formulas for course and distance giveλ12 = -166.6°,[3] α1 = -94.41°,α2 = -78.42°, andσ12 = 168.56°. Taking the earth radius to beR = 6371 km, the distance iss12 = 18743 km.
To compute points along the route, first findα0 = -56.74°,σ01 = -96.76°,σ02 = 71.8°,λ01 = 98.07°, andλ0 = -169.67°.Then to compute the midpoint of the route (for example), takeσ = (σ01 + σ02) = -12.48°, and solveforφ = -6.81°,λ = -159.18°, andα = -57.36°.
If the geodesic is computed accurately on the WGS84 ellipsoid,[4] the resultsare α1 = -94.82°, α2 = -78.29°, ands12 = 18752 km. The midpoint of the geodesic isφ = -7.07°, λ = -159.31°,α = -57.45°.
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.