Great-circle navigation

File:Sphere geodesic 4sigma.svg

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

:$\backslash begin\{align\}$

\tan\alpha_1&=\frac{\cos\phi_2\sin\lambda_{12}}{ \cos\phi_1\sin\phi_2-\sin\phi_1\cos\phi_2\cos\lambda_{12}},\\

\tan\alpha_2&=\frac{\cos\phi_1\sin\lambda_{12}}{-\cos\phi_2\sin\phi_1+\sin\phi_2\cos\phi_1\cos\lambda_{12}},\\

\end{align}

where λ₁₂ = λ₂ − λ₁In 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 α₁,α₂ 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, σ₁₂, is given by

:$\backslash tan\backslash sigma\_\{12\}=\backslash frac\{\backslash sqrt\{(\backslash cos\backslash phi\_1\backslash sin\backslash phi\_2-\backslash sin\backslash phi\_1\backslash cos\backslash phi\_2\backslash cos\backslash lambda\_\{12\})^2\; +\; (\backslash cos\backslash phi\_2\backslash sin\backslash lambda\_\{12\})^2\}\}\{\backslash sin\backslash phi\_1\backslash sin\backslash phi\_2+\backslash cos\backslash phi\_1\backslash cos\backslash phi\_2\backslash cos\backslash lambda\_\{12\}\}.${{refn|group=note|A simpler formula is

:$$

\cos\sigma_{12}=\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\lambda_{12};

however, this is numerically less accurate if σ₁₂ small.}}{{refn|group=note|These equations for α₁,α₂,σ₁₂ are suitable for implementation

on modern calculators and computers. For hand computations with logarithms,

Delambre's analogies{{cite book

|last = Todhunter

|first = I.

|authorlink = Isaac Todhunter

|title = Spherical Trigonometry

|year = 1871

|publisher = MacMillan

|edition = 3rd

|page = [https://archive.org/details/sphericaltrigon05todhgoog/page/n40 26]

|url = https://archive.org/details/sphericaltrigon05todhgoog}}

were usually used:

:$$

\begin{align}

\cos\tfrac12(\alpha_2+\alpha_1) \sin\tfrac12\sigma_{12} &= \sin\tfrac12(\phi_2-\phi_1) \cos\tfrac12\lambda_{12},\\

\sin\tfrac12(\alpha_2+\alpha_1) \sin\tfrac12\sigma_{12} &= \cos\tfrac12(\phi_2+\phi_1) \sin\tfrac12\lambda_{12},\\

\cos\tfrac12(\alpha_2-\alpha_1) \cos\tfrac12\sigma_{12} &= \cos\tfrac12(\phi_2-\phi_1) \cos\tfrac12\lambda_{12},\\

\sin\tfrac12(\alpha_2-\alpha_1) \cos\tfrac12\sigma_{12} &= \sin\tfrac12(\phi_2+\phi_1) \sin\tfrac12\lambda_{12}.

\end{align}

McCaw{{cite journal

|last = McCaw

|first = G. T.

|title = Long lines on the Earth

|journal = Empire Survey Review

|volume = 1

|number = 6

|pages = 259–263

|doi = 10.1179/sre.1932.1.6.259

|year = 1932}} 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 {{!}}λ₁₂{{!}} ≤ π (otherwise the longer path is found).}}

(The numerator of this formula contains the quantities that were used to determine

tan α₁.)

The distance along the great circle will then be s₁₂ = Rσ₁₂, where R is the assumed radius

of the Earth and σ₁₂ is expressed in radians.

Using the mean Earth radius, R = R₁ ≈ {{convert|6371|km|mi|abbr=on}} yields results for

the distance s₁₂ which are within 1% of the geodesic length for the WGS84 ellipsoid; see Geodesics on an ellipsoid for details.

{{cleanup merge|Position angle|21=section}}

File:PositionAngle.svg

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 {{math|φ_s}} and geodetic longitude {{math|λ_s}}, where {{math|φ}} is considered positive if north of the equator, and where {{math|λ}} is considered positive if east of Greenwich. In the geocentric coordinate system centered at the center of the sphere, the Cartesian components are

::$\{\backslash mathbf\; s\}=R\backslash left(\backslash begin\{array\}\{c\}\; \backslash cos\backslash varphi\_s\; \backslash cos\backslash lambda\_s\; \backslash \backslash$

\cos\varphi_s \sin\lambda_s \\

\sin\varphi_s

\end{array}\right)

and the target position is

::$\{\backslash mathbf\; t\}=R\backslash left(\backslash begin\{array\}\{c\}\; \backslash cos\backslash varphi\_t\; \backslash cos\backslash lambda\_t\; \backslash \backslash$

\cos\varphi_t \sin\lambda_t \\

\sin\varphi_t

\end{array}\right).

The North Pole is at

::$\{\backslash mathbf\; N\}=R\backslash left(\backslash begin\{array\}\{c\}\; 0\; \backslash \backslash$

0 \\

1

\end{array}\right).

The minimum distance {{math|d}} is the distance along a great circle that runs through {{math|s}} and {{math|t}}. It is calculated in a plane that contains the sphere center and the great circle,

::$d\_\{s,t\}=R\backslash theta\_\{s,t\}$

where {{math|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

::$\backslash mathbf\{s\}\backslash cdot\; \backslash mathbf\{t\}\; =\; R^2\backslash cos\; \backslash theta\_\{s,t\}\; =\; R^2(\backslash sin\backslash varphi\_s\backslash sin\backslash varphi\_t+\backslash cos\backslash varphi\_s\backslash cos\backslash varphi\_t\backslash cos(\backslash lambda\_t-\backslash lambda\_s))$

If the ship steers straight to the North Pole, the travel distance is

::$d\_\{s,N\}\; =\; R\backslash theta\_\{s,N\}\; =\; R(\backslash pi/2-\backslash varphi\_s)$

If a ship starts at {{math|t}} and sails straight to the North Pole, the travel distance is

::$d\_\{t,N\}\; =\; R\backslash theta\_\{t,n\}\; =R(\backslash pi/2-\backslash varphi\_t)$

The cosine formula of spherical trigonometry{{AS ref|4.3.149}} yields for the

angle {{math|p}} between the great circles through {{math|s}} that point to the North on one hand and to {{math|t}} on the other hand

::$\backslash cos\backslash theta\_\{t,N\}\; =\; \backslash cos\backslash theta\_\{s,t\}\backslash cos\backslash theta\_\{s,N\}+\backslash sin\backslash theta\_\{s,t\}\backslash sin\backslash theta\_\{s,N\}\backslash cos\; p.$

::$\backslash sin\backslash varphi\_t\; =\; \backslash cos\backslash theta\_\{s,t\}\backslash sin\backslash varphi\_s\; +\backslash sin\backslash theta\_\{s,t\}\backslash cos\backslash varphi\_s\backslash cos\; p.$

The sine formula yields

::$\backslash frac\{\backslash sin\; p\}\{\backslash sin\; \backslash theta\_\{t,N\}\}\; =\; \backslash frac\{\backslash sin(\backslash lambda\_t-\backslash lambda\_s)\}\{\backslash sin\backslash theta\_\{s,t\}\}.$

Solving this for {{math|sin θ_s,t}} and insertion in the previous formula gives an expression for the tangent of the position angle,

::$\backslash sin\backslash varphi\_t\; =\; \backslash cos\backslash theta\_\{s,t\}\backslash sin\backslash varphi\_s\; +\backslash frac\{\backslash sin(\backslash lambda\_t-\backslash lambda\_s)\}\{\backslash sin\; p\}\backslash cos\backslash varphi\_t\backslash cos\backslash varphi\_s\backslash cos\; p;$

::$\backslash tan\; p\; =\; \backslash frac\{\backslash sin(\backslash lambda\_t-\backslash lambda\_s)\backslash cos\backslash varphi\_t\backslash cos\backslash varphi\_s\}\{\backslash sin\backslash varphi\_t-\backslash cos\backslash theta\_\{s,t\}\backslash sin\backslash varphi\_s\}.$

Because the brief derivation gives an angle between 0 and {{math|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 {{math|p}} such that use of the correct branch of the inverse tangent allows to produce an angle in the full range {{math|-π≤p≤π}}.

The computation starts from a construction of the great circle between {{math|s}} and {{math|t}}. It lies in the plane that contains the sphere center, {{math|s}} and {{math|t}} and is constructed rotating {{math|s}} by the angle {{math|θ_s,t}} around an axis {{math|ω}}. The axis is perpendicular to the plane of the great circle and computed by the normalized vector cross product of the two positions:

::$\backslash mathbf\{\backslash omega\}\; =\; \backslash frac\{1\}\{R^2\backslash sin\; \backslash theta\_\{s,t\}\}\backslash mathbf\{s\}\backslash times\; \backslash mathbf\{t\}$

= \frac{1}{\sin \theta_{s,t}}\left(\begin{array}{c}

\cos\varphi_s\sin\lambda_s\sin\varphi_t -\sin\varphi_s\cos\varphi_t\sin\lambda_t

\\

\sin\varphi_s\cos\lambda_t\cos\varphi_t -\cos\varphi_s\sin\varphi_t\cos\lambda_s

\\

\cos\varphi_s\cos\varphi_t\sin(\lambda_t-\lambda_s)

\end{array}\right).

A right-handed tilted coordinate system with the center at the center of the sphere is given by the

following three axes: the

axis {{math|s}}, the axis

::$\backslash mathbf\{s\}\_\backslash perp\; =\; \backslash omega\; \backslash times\; \backslash frac\{1\}\{R\}\backslash mathbf\{s\}\; =$

\frac{1}{\sin\theta_{s,t}}

\left(\begin{array}{c}

\cos\varphi_t\cos\lambda_t(\sin^2\varphi_s+\cos^2\varphi_s\sin^2\lambda_s)-\cos\lambda_s(\sin\varphi_s\cos\varphi_s\sin\varphi_t+\cos^2\varphi_s\sin\lambda_s\cos\varphi_t\sin\lambda_t)\\

\cos\varphi_t\sin\lambda_t(\sin^2\varphi_s+\cos^2\varphi_s\cos^2\lambda_s)-\sin\lambda_s(\sin\varphi_s\cos\varphi_s\sin\varphi_t+\cos^2\varphi_s\cos\lambda_s\cos\varphi_t\cos\lambda_t)\\

\cos\varphi_s[\cos\varphi_s\sin\varphi_t-\sin\varphi_s\cos\varphi_t\cos(\lambda_t-\lambda_s)]

\end{array}\right)

and the axis {{math|ω}}.

A position along the great circle is

::$\backslash mathbf\{s\}(\backslash theta)\; =\; \backslash cos\backslash theta\; \backslash mathbf\{s\}+\backslash sin\backslash theta\; \backslash mathbf\{s\}\_\backslash perp,\backslash quad\; 0\backslash le\backslash theta\backslash le\; 2\backslash pi.$

The compass direction is given by inserting the two vectors {{math|s}} and {{math|s_⊥}} and computing the gradient of the vector with respect to {{math|θ}} at {{math|θ{{=}}0}}.

::$\backslash frac\{\backslash partial\}\{\backslash partial\backslash theta\}\backslash mathbf\{s\}\_\{\backslash mid\; \backslash theta=0\}=\backslash mathbf\{s\}\_\backslash perp.$

The angle {{math|p}} is given by splitting this direction along two orthogonal directions in the plane tangential to the sphere at the point {{math|s}}. The two directions are given by the partial derivatives of {{math|s}} with respect to {{math|φ}} and with respect to {{math|λ}}, normalized to unit length:

::$\backslash mathbf\{u\}\_N\; =\; \backslash left($

\begin{array}{c}

-\sin\varphi_s\cos\lambda_s\\

-\sin\varphi_s\sin\lambda_s\\

\cos\varphi_s

\end{array}\right);

::$\backslash mathbf\{u\}\_E\; =\; \backslash left(\backslash begin\{array\}\{c\}$

-\sin\lambda_s\\

\cos\lambda_s\\

0

\end{array}

\right);

::$\backslash mathbf\{u\}\_N\backslash cdot\; \backslash mathbf\{s\}\; =\; \backslash mathbf\{u\}\_E\backslash cdot\; \backslash mathbf\{u\}\_N\; =0$

{{math|u_N}} points north and {{math|u_E}} points east at the position {{math|s}}.

The position angle {{math|p}} projects {{math|s_⊥}}

into these two directions,

::$\backslash mathbf\{s\}\_\backslash perp\; =\; \backslash cos\; p\; \backslash ,\backslash mathbf\{u\}\_N+\backslash sin\; p\backslash ,\; \backslash mathbf\{u\}\_E$,

where the positive sign means the positive position angles are defined to be north over east. The values of the cosine and sine of {{math|p}} are computed by multiplying this equation on both sides with the two unit vectors,

::$\backslash cos\; p\; =\; \backslash mathbf\{s\}\_\backslash perp\; \backslash cdot\; \backslash mathbf\{u\}\_N$

=\frac{1}{\sin\theta_{s,t}}[\cos\varphi_s\sin\varphi_t - \sin\varphi_s\cos\varphi_t\cos(\lambda_t-\lambda_s)];

::$\backslash sin\; p\; =\; \backslash mathbf\{s\}\_\backslash perp\; \backslash cdot\; \backslash mathbf\{u\}\_E$

=\frac{1}{\sin\theta_{s,t}}[\cos\varphi_t\sin(\lambda_t-\lambda_s)].

Instead of inserting the convoluted expression of {{math|s_⊥}}, the evaluation may employ that the triple product is invariant under a circular shift

of the arguments:

::$\backslash cos\; p\; =\; (\backslash mathbf\{\backslash omega\}\backslash times\; \backslash frac\{1\}\{R\}\backslash mathbf\{s\})\backslash cdot\; \backslash mathbf\{u\}\_N$

= \omega\cdot(\frac{1}{R}\mathbf{s}\times \mathbf{u}_N).

If atan2 is used to compute the value, one can reduce both expressions by division through {{math|cos φ_t}}

and multiplication by {{math|sin θ_s,t}},

because these values are always positive and that operation does not change signs; then effectively

::$\backslash tan\; p\; =\; \backslash frac\{\backslash sin(\backslash lambda\_t-\backslash lambda\_s)\}\{\backslash cos\backslash varphi\_s\backslash tan\backslash varphi\_t\; -\backslash sin\backslash varphi\_s\backslash cos(\backslash lambda\_t-\backslash lambda\_s)\}.$

To find the way-points, that is the positions of selected points on the great circle between

P₁ and P₂, 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 λ₀ — see Fig 1. The azimuth at this point, α₀, is given by

:$\backslash tan\backslash alpha\_0\; =\; \backslash frac$

{\sin\alpha_1 \cos\phi_1}{\sqrt{\cos^2\alpha_1 + \sin^2\alpha_1\sin^2\phi_1}}.{{refn|group=note|A simpler formula is

:$$

\sin\alpha_0 = \sin\alpha_1 \cos\phi_1;

however, this is less accurate

α₀ ≈ ±{{frac|1|2}}π.

}}

Let the angular distances along the great circle from A to P₁ and P₂ be σ₀₁ and σ₀₂ respectively. Then using Napier's rules we have

:$$

\tan\sigma_{01} = \frac{\tan\phi_1}{\cos\alpha_1}

\qquad(If φ₁ = 0 and α₁ = {{frac|1|2}}π, use σ₀₁ = 0).

This gives σ₀₁, whence σ₀₂ = σ₀₁ + σ₁₂.

The longitude at the node is found from

:$$

\begin{align}

\tan\lambda_{01} &= \frac{\sin\alpha_0\sin\sigma_{01}}{\cos\sigma_{01}},\\

\lambda_0 &= \lambda_1 - \lambda_{01}.

\end{align}

File:Sphere geodesic 2sigma.svg

Finally, calculate the position and azimuth at an arbitrary point, P (see Fig. 2), by the spherical version of the direct geodesic problem.{{refn|group=note|The direct geodesic problem, finding the position of P₂ given P₁, α₁,

and s₁₂, can also be solved by

formulas for solving a spherical triangle, as follows,

:$$

\begin{align}

\sigma_{12} &= s_{12}/R,\\

\sin\phi_2 &= \sin\phi_1\cos\sigma_{12} + \cos\phi_1\sin\sigma_{12}\cos\alpha_1,\quad\text{or}\\

\tan\phi_2 &= \frac{\sin\phi_1\cos\sigma_{12} + \cos\phi_1\sin\sigma_{12}\cos\alpha_1}

{\sqrt{ (\cos\phi_1\cos\sigma_{12} - \sin\phi_1\sin\sigma_{12}\cos\alpha_1)^2 + (\sin\sigma_{12}\sin\alpha_1)^2 }},\\

\tan\lambda_{12} &= \frac{\sin\sigma_{12}\sin\alpha_1}

{\cos\phi_1\cos\sigma_{12} - \sin\phi_1\sin\sigma_{12}\cos\alpha_1},\\

\lambda_2 &= \lambda_1 + \lambda_{12},\\

\tan\alpha_2 &= \frac{\sin\alpha_1}

{\cos\sigma_{12}\cos\alpha_1 - \tan\phi_1\sin\sigma_{12}}.

\end{align}

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.}} Napier's rules give

:$$

{\color{white}.\,\qquad)}\tan\phi = \frac

{\cos\alpha_0\sin\sigma}{\sqrt{\cos^2\sigma + \sin^2\alpha_0\sin^2\sigma}},{{refn|group=note|A simpler formula is

:$$

\sin\phi = \cos\alpha_0\sin\sigma;

however, this is less accurate when

φ ≈ ±{{frac|1|2}}π}}

:$$

\begin{align}

\tan(\lambda - \lambda_0) &= \frac

{\sin\alpha_0\sin\sigma}{\cos\sigma},\\

\tan\alpha &= \frac

{\tan\alpha_0}{\cos\sigma}.

\end{align}

The atan2 function should be used to determine

σ₀₁,

λ, and α.

For example, to find the

midpoint of the path, substitute σ = {{frac|1|2}}(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 = +{{frac|1|2}}π.

It may be convenient to parameterize the route in terms of the longitude using

:$\backslash tan\backslash phi\; =\; \backslash cot\backslash alpha\_0\backslash sin(\backslash lambda-\backslash lambda\_0).${{refn|group=note|

The following is used: $\backslash cos\backslash sigma\; =\; \backslash cos\backslash phi\; \backslash cos(\backslash lambda-\backslash lambda\_0)$}}

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 $(\backslash phi,\backslash lambda)$

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.

Compute the great circle route from Valparaíso,

φ₁ = −33°,

λ₁ = −71.6°, to

Shanghai,

φ₂ = 31.4°,

λ₂ = 121.8°.

The formulas for course and distance give

λ₁₂ = −166.6°,λ₁₂

is reduced to the range [−180°, 180°] by adding or subtracting 360° as

necessary

α₁ = −94.41°,

α₂ = −78.42°, and

σ₁₂ = 168.56°. Taking the earth radius to be

R = 6371 km, the distance is

s₁₂ = 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 = {{frac|1|2}}(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 ellipsoid,

{{cite journal

|first=C. F. F.

|last=Karney

|title=Algorithms for geodesics

|journal=Journal of Geodesy

|volume = 87

|number = 1

|year = 2013

|pages = 43–55

|doi=10.1007/s00190-012-0578-z

|arxiv=1109.4448

|bibcode=2013JGeod..87...43K

|doi-access=free

}}

the results

are α₁ = −94.82°, α₂ = −78.29°, and

s₁₂ = 18752 km. The midpoint of the geodesic is

φ = −7.07°, λ = −159.31°,

α = −57.45°.

File:Admiralty Chart No 132 Gnomonic Chart of Indian and Southern Oceans, Published 1914.jpg

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.

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

{{reflist|group=note}}

{{reflist}}

• [http://mathworld.wolfram.com/GreatCircle.html Great Circle – from MathWorld] Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
• [https://www.greatcirclemap.com Great Circle Map] Interactive tool for plotting great circle routes on a sphere.
• [http://www.gcmap.com/ Great Circle Mapper] Interactive tool for plotting great circle routes.
• [http://edwilliams.org/gccalc.htm Great Circle Calculator] deriving (initial) course and distance between two points.
• [http://www.acscdg.com/ Great Circle Distance] Graphical tool for drawing great circles over maps. Also shows distance and azimuth in a table.
• [http://sites.google.com/site/navigationalalgorithms/ Google assistance program for orthodromic navigation]

Category:Navigation

Category:Circles

Category:Spherical curves