View source for Great-circle navigation

{{Short description|Flight or sailing route along the shortest path between two points on a globe's surface}}
{{for|the navigation on an ellipsoid|Geodesics on an ellipsoid}}
[[File: Grosskreis.jpg |thumb|230px | Orthodromic course drawn on the Earth [[globe]]]]

'''Great-circle navigation''' or '''orthodromic navigation''' (related to '''orthodromic course'''; {{etymology|grc|''{{Wikt-lang|grc|ορθός}}'' ({{grc-transl|ορθός}})|right angle||''{{Wikt-lang|grc|δρόμος}}'' ({{grc-transl|δρόμος}})|path}}) is the practice of [[navigation|navigating]] a vessel (a [[ship]] or [[aircraft]]) along a [[great circle]]. Such routes yield the shortest [[distance]]  between two points on the globe.<ref name="WeintritNeumann2011">{{cite book |author1=Adam Weintrit |author2=Tomasz Neumann |title=Methods and Algorithms in Navigation: Marine Navigation and Safety of Sea Transportation |url=https://books.google.com/books?id=buMsPGyE7boC&q=loxodromic+navigation&pg=PA139 |date=7 June 2011 |publisher=[[CRC Press]] |isbn=978-0-415-69114-7 |pages=139–}}</ref>

==Course==
[[File:Sphere geodesic 4sigma.svg|thumb|200px|right|Figure 1. The great circle path between (&phi;<sub>1</sub>,&nbsp;&lambda;<sub>1</sub>) and (&phi;<sub>2</sub>,&nbsp;&lambda;<sub>2</sub>).]]

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''<sub>1</sub>&nbsp;=&nbsp;(&phi;<sub>1</sub>,&lambda;<sub>1</sub>) and plans to travel the great circle to a point at point ''P''<sub>2</sub>&nbsp;=&nbsp;(&phi;<sub>2</sub>,&lambda;<sub>2</sub>) (see Fig. 1, &phi; is the latitude, positive northward, and &lambda; is the longitude, positive eastward), the initial and final courses &alpha;<sub>1</sub> and &alpha;<sub>2</sub> are given by [[Solution of triangles#Two sides and the included angle given|formulas for solving a spherical triangle]]
:<math>\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}</math>
where &lambda;<sub>12</sub>&nbsp;=&nbsp;&lambda;<sub>2</sub>&nbsp;&minus;&nbsp;&lambda;<sub>1</sub><ref group=note>In the article on [[great-circle distance]]s,
the notation &Delta;&lambda;&nbsp;=&nbsp;&lambda;<sub>12</sub>
and &Delta;&sigma;&nbsp;=&nbsp;&sigma;<sub>12</sub> is used.  The notation in this article is needed to
deal with differences between other points, e.g., &lambda;<sub>01</sub>.</ref>
and the quadrants of &alpha;<sub>1</sub>,&alpha;<sub>2</sub> 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, &sigma;<sub>12</sub>, is given by
:<math>\tan\sigma_{12}=\frac{\sqrt{(\cos\phi_1\sin\phi_2-\sin\phi_1\cos\phi_2\cos\lambda_{12})^2 + (\cos\phi_2\sin\lambda_{12})^2}}{\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\lambda_{12}}.</math>{{refn|group=note|A simpler formula is
:<math>
\cos\sigma_{12}=\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\lambda_{12};
</math>
however, this is numerically less accurate if &sigma;<sub>12</sub> small.}}{{refn|group=note|These equations for &alpha;<sub>1</sub>,&alpha;<sub>2</sub>,&sigma;<sub>12</sub> are suitable for implementation
on modern calculators and computers.  For hand computations with logarithms,
[[Delambre]]'s analogies<ref>{{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}}
</ref> were usually used:
:<math>
\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}
</math>
McCaw<ref>{{cite journal
|last = McCaw
|first = G. T.
|title = Long lines on the Earth
|journal = Empire Survey Review
|volume = 1
|number = 6
|pages = 259&ndash;263
|doi = 10.1179/sre.1932.1.6.259
|year = 1932}}</ref> 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 {{!}}&lambda;<sub>12</sub>{{!}}&nbsp;&le;&nbsp;&pi; (otherwise the longer path is found).}}
(The numerator of this formula contains the quantities that were used to determine
tan&nbsp;&alpha;<sub>1</sub>.)
The distance along the great circle will then be ''s''<sub>12</sub>&nbsp;=&nbsp;''R''&sigma;<sub>12</sub>, where ''R'' is the assumed radius
of the Earth and &sigma;<sub>12</sub> is expressed in [[Radian#Conversions|radians]].
Using the [[Earth radius#Mean radius|mean Earth radius]], ''R''&nbsp;=&nbsp;''R''<sub>1</sub>&nbsp;&asymp;&nbsp;{{convert|6371|km|mi|abbr=on}}  yields results for
the distance ''s''<sub>12</sub> which are within 1% of the [[geodesic length]] for the [[WGS84]] ellipsoid; see [[Geodesics on an ellipsoid]] for details.

===Relation to geocentric coordinate system===
{{cleanup merge|Position angle|21=section}}

[[File:PositionAngle.svg|thumb|400px|The position angle of the point '''t''' at the point '''s''' is the angle at which the green and the dashed great circles intersect at '''s'''. The unit directions {{math|u<sub>E</sub>}}, {{math|u<sub>N</sub>}} and the rotation axis {{math|&omega;}} are marked by arrows.]]
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<sub>s</sub>}} and geodetic [[longitude]] {{math|λ<sub>s</sub>}}, where {{math|φ}} is considered positive if north of the equator, and where {{math|λ}} is considered positive if east of [[Prime meridian|Greenwich]]. In the [[geocentric coordinate system]] centered at the center of the sphere, the Cartesian components are

::<math>{\mathbf s}=R\left(\begin{array}{c} \cos\varphi_s \cos\lambda_s \\
\cos\varphi_s \sin\lambda_s \\
\sin\varphi_s
\end{array}\right) </math>
and the target position is

::<math>{\mathbf t}=R\left(\begin{array}{c} \cos\varphi_t \cos\lambda_t \\
\cos\varphi_t \sin\lambda_t \\
\sin\varphi_t
\end{array}\right). </math>
The North Pole is at
::<math>{\mathbf N}=R\left(\begin{array}{c} 0 \\
0 \\
1
\end{array}\right). </math>
The [[Great-circle distance|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]],
::<math> d_{s,t}=R\theta_{s,t}</math>
where {{math|θ}} is the angular distance of two points viewed from the center of the sphere, measured in [[radian]]s. The cosine of the angle is calculated by the [[dot product]]  of the two vectors
::<math>\mathbf{s}\cdot \mathbf{t} = R^2\cos \theta_{s,t} = R^2(\sin\varphi_s\sin\varphi_t+\cos\varphi_s\cos\varphi_t\cos(\lambda_t-\lambda_s))</math>
If the ship steers straight to the North Pole, the travel distance is
::<math>d_{s,N} = R\theta_{s,N} = R(\pi/2-\varphi_s)</math>
If a ship starts at {{math|'''t'''}} and sails straight to the North Pole, the travel distance is
::<math>d_{t,N} = R\theta_{t,n} =R(\pi/2-\varphi_t)</math>

====Derivation====
The ''cosine formula'' of [[spherical trigonometry]]<ref name=AS>{{AS ref|4.3.149}}</ref> 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
::<math>\cos\theta_{t,N} = \cos\theta_{s,t}\cos\theta_{s,N}+\sin\theta_{s,t}\sin\theta_{s,N}\cos p.</math>
::<math>\sin\varphi_t = \cos\theta_{s,t}\sin\varphi_s +\sin\theta_{s,t}\cos\varphi_s\cos p.</math>
The ''sine formula'' yields
::<math>\frac{\sin p}{\sin \theta_{t,N}} = \frac{\sin(\lambda_t-\lambda_s)}{\sin\theta_{s,t}}.</math>
Solving this for {{math|sin θ<sub>s,t</sub>}} and insertion in the previous formula gives an expression for the tangent of the position angle,
::<math>\sin\varphi_t = \cos\theta_{s,t}\sin\varphi_s +\frac{\sin(\lambda_t-\lambda_s)}{\sin p}\cos\varphi_t\cos\varphi_s\cos p;</math>
::<math>\tan p = \frac{\sin(\lambda_t-\lambda_s)\cos\varphi_t\cos\varphi_s}{\sin\varphi_t-\cos\theta_{s,t}\sin\varphi_s}.</math>

====Further details====
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 [[Atan2|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<sub>s,t</sub>}} 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:
::<math>\mathbf{\omega} = \frac{1}{R^2\sin \theta_{s,t}}\mathbf{s}\times \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).
</math>
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
::<math>\mathbf{s}_\perp = \omega \times \frac{1}{R}\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)
</math>
and the axis {{math|ω}}.
A position along the great circle is
::<math>\mathbf{s}(\theta) = \cos\theta \mathbf{s}+\sin\theta \mathbf{s}_\perp,\quad 0\le\theta\le 2\pi.</math>
The compass direction is given by inserting the two vectors {{math|'''s'''}} and {{math|'''s'''<sub>&perp;</sub>}} and computing the gradient of the vector with respect to {{math|θ}} at {{math|θ{{=}}0}}.
::<math>\frac{\partial}{\partial\theta}\mathbf{s}_{\mid \theta=0}=\mathbf{s}_\perp.</math>
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|&phi;}} and with respect to {{math|&lambda;}}, normalized to unit length:
::<math>\mathbf{u}_N = \left(
\begin{array}{c}
-\sin\varphi_s\cos\lambda_s\\
-\sin\varphi_s\sin\lambda_s\\
\cos\varphi_s
\end{array}\right);</math>
::<math>\mathbf{u}_E = \left(\begin{array}{c}
-\sin\lambda_s\\
\cos\lambda_s\\
0
\end{array}
\right);</math>
::<math>\mathbf{u}_N\cdot \mathbf{s} = \mathbf{u}_E\cdot \mathbf{u}_N =0 </math>
{{math|'''u'''<sub>N</sub>}} points north and {{math|'''u'''<sub>E</sub>}} points east at the position {{math|'''s'''}}.
The position angle {{math|p}} projects {{math|'''s'''<sub>&perp;</sub>}}
into these two directions,
::<math>\mathbf{s}_\perp = \cos p \,\mathbf{u}_N+\sin p\, \mathbf{u}_E</math>,
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,
::<math>\cos p = \mathbf{s}_\perp \cdot \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)];
</math>
::<math>\sin p = \mathbf{s}_\perp \cdot \mathbf{u}_E
=\frac{1}{\sin\theta_{s,t}}[\cos\varphi_t\sin(\lambda_t-\lambda_s)].
</math>
Instead of inserting the convoluted expression of {{math|'''s'''<sub>&perp;</sub>}}, the evaluation may employ that the [[Triple product#Scalar triple product|triple product]] is invariant under a circular shift
of the arguments:
::<math>\cos p = (\mathbf{\omega}\times \frac{1}{R}\mathbf{s})\cdot \mathbf{u}_N
= \omega\cdot(\frac{1}{R}\mathbf{s}\times \mathbf{u}_N).</math>

If [[atan2]] is used to compute the value, one can reduce both expressions by division through {{math|cos &phi;<sub>t</sub>}}
and multiplication by {{math|sin &theta;<sub>s,t</sub>}},
because these values are always positive and that operation does not change signs; then effectively
::<math>\tan p = \frac{\sin(\lambda_t-\lambda_s)}{\cos\varphi_s\tan\varphi_t -\sin\varphi_s\cos(\lambda_t-\lambda_s)}.</math>

==Finding way-points==

To find the [[way-point]]s, that is the positions of selected points on the great circle between
''P''<sub>1</sub> and ''P''<sub>2</sub>, we first extrapolate the great circle back to its ''[[Orbital node|node]]'' ''A'', the point
at which the great circle crosses the
equator in the northward direction: let the longitude of this point be &lambda;<sub>0</sub> — see Fig&nbsp;1.  The [[azimuth]] at this point, &alpha;<sub>0</sub>, is given by
:<math>\tan\alpha_0 = \frac
{\sin\alpha_1 \cos\phi_1}{\sqrt{\cos^2\alpha_1 + \sin^2\alpha_1\sin^2\phi_1}}.</math>{{refn|group=note|A simpler formula is
:<math>
\sin\alpha_0 = \sin\alpha_1 \cos\phi_1;
</math>
however, this is less accurate
&alpha;<sub>0</sub>&nbsp;&asymp;&nbsp;&plusmn;{{frac|1|2}}&pi;.
}}
Let the angular distances along the great circle from ''A''  to ''P''<sub>1</sub> and ''P''<sub>2</sub> be &sigma;<sub>01</sub> and &sigma;<sub>02</sub> respectively. Then using [[Spherical trigonometry#Napier's rules for quadrantal triangles|Napier's rules]] we have
:<math>
\tan\sigma_{01} = \frac{\tan\phi_1}{\cos\alpha_1}
\qquad</math>(If &phi;<sub>1</sub>&nbsp;=&nbsp;0 and &alpha;<sub>1</sub>&nbsp;=&nbsp;{{frac|1|2}}&pi;, use &sigma;<sub>01</sub>&nbsp;=&nbsp;0).

This gives &sigma;<sub>01</sub>, whence &sigma;<sub>02</sub>&nbsp;=&nbsp;&sigma;<sub>01</sub>&nbsp;+&nbsp;&sigma;<sub>12</sub>.

The longitude at the node is found from
<!-- Don't simplify sin(sigma)/cos(sigma) to tan(sigma).  This gives the wrong quadrant for lambda. -->
:<math>
\begin{align}
\tan\lambda_{01} &= \frac{\sin\alpha_0\sin\sigma_{01}}{\cos\sigma_{01}},\\
\lambda_0 &= \lambda_1 - \lambda_{01}.
\end{align}
</math>
[[File:Sphere geodesic 2sigma.svg|thumb|200px|right|Figure 2. The great circle path between a node (an equator crossing) and an arbitrary point (&phi;,&lambda;).]]
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 ''P''<sub>2</sub> given ''P''<sub>1</sub>, &alpha;<sub>1</sub>,
and ''s''<sub>12</sub>, can also be solved by
[[Solution of triangles#Two sides and the included angle given|formulas for solving a spherical triangle]], as follows,
:<math>
\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}
</math>
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 &sigma;
(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
:<math>
{\color{white}.\,\qquad)}\tan\phi = \frac
{\cos\alpha_0\sin\sigma}{\sqrt{\cos^2\sigma + \sin^2\alpha_0\sin^2\sigma}},</math>{{refn|group=note|A simpler formula is
:<math>
\sin\phi = \cos\alpha_0\sin\sigma;</math>
however, this is less accurate when
&phi;&nbsp;&asymp;&nbsp;&plusmn;{{frac|1|2}}&pi;}}
:<!-- Don't simplify sin(sigma)/cos(sigma) to tan(sigma).  This gives the wrong quadrant for lambda. --><math>
\begin{align}
\tan(\lambda - \lambda_0) &= \frac
{\sin\alpha_0\sin\sigma}{\cos\sigma},\\
\tan\alpha &= \frac
{\tan\alpha_0}{\cos\sigma}.
\end{align}
</math>
The [[atan2]] function should be used to determine
&sigma;<sub>01</sub>,
&lambda;, and &alpha;.
For example, to find the
midpoint of the path, substitute &sigma;&nbsp;=&nbsp;{{frac|1|2}}(&sigma;<sub>01</sub>&nbsp;+&nbsp;&sigma;<sub>02</sub>); alternatively
to find the point a distance ''d'' from the starting point, take &sigma;&nbsp;=&nbsp;&sigma;<sub>01</sub>&nbsp;+&nbsp;''d''/''R''.
Likewise, the ''vertex'', the point on the great
circle with greatest latitude, is found by substituting &sigma;&nbsp;=&nbsp;+{{frac|1|2}}&pi;.
It may be convenient to parameterize the route in terms of the longitude using
:<math>\tan\phi = \cot\alpha_0\sin(\lambda-\lambda_0).</math>{{refn|group=note|
The following is used: <math>\cos\sigma = \cos\phi \cos(\lambda-\lambda_0)</math>}}
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]]s.  The path determined in this way
gives the [[great ellipse]] joining the end points, provided the coordinates <math>(\phi,\lambda)</math>
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]],
&phi;<sub>1</sub>&nbsp;=&nbsp;&minus;33°,
&lambda;<sub>1</sub>&nbsp;=&nbsp;&minus;71.6°, to
[[Shanghai]],
&phi;<sub>2</sub>&nbsp;=&nbsp;31.4°,
&lambda;<sub>2</sub>&nbsp;=&nbsp;121.8°.

The formulas for course and distance give
&lambda;<sub>12</sub>&nbsp;=&nbsp;&minus;166.6°,<ref group=note>&lambda;<sub>12</sub>
is reduced to the range [&minus;180°,&nbsp;180°] by adding or subtracting 360° as
necessary</ref>
&alpha;<sub>1</sub>&nbsp;=&nbsp;&minus;94.41°,
&alpha;<sub>2</sub>&nbsp;=&nbsp;&minus;78.42°, and
&sigma;<sub>12</sub>&nbsp;=&nbsp;168.56°.  Taking the [[Earth radius#Mean radius|earth radius]] to be
''R''&nbsp;=&nbsp;6371&nbsp;km, the distance is
''s''<sub>12</sub>&nbsp;=&nbsp;18743&nbsp;km.

To compute points along the route, first find
&alpha;<sub>0</sub>&nbsp;=&nbsp;&minus;56.74°,
&sigma;<sub>01</sub>&nbsp;=&nbsp;&minus;96.76°,
&sigma;<sub>02</sub>&nbsp;=&nbsp;71.8°,
&lambda;<sub>01</sub>&nbsp;=&nbsp;98.07°, and
&lambda;<sub>0</sub>&nbsp;=&nbsp;&minus;169.67°.
Then to compute the midpoint of the route (for example), take
&sigma;&nbsp;=&nbsp;{{frac|1|2}}(&sigma;<sub>01</sub>&nbsp;+&nbsp;&sigma;<sub>02</sub>)&nbsp;=&nbsp;&minus;12.48°, and solve
for
&phi;&nbsp;=&nbsp;&minus;6.81°,
&lambda;&nbsp;=&nbsp;&minus;159.18°, and
&alpha;&nbsp;=&nbsp;&minus;57.36°.

If the geodesic is computed accurately on the [[WGS84]] ellipsoid,<ref>
{{cite journal
 |first=C. F. F.
 |last=Karney
 |title=Algorithms for geodesics
 |journal=Journal of Geodesy
 |volume = 87
 |number = 1
 |year = 2013
 |pages = 43&ndash;55
 |doi=10.1007/s00190-012-0578-z
 |arxiv=1109.4448
 |bibcode=2013JGeod..87...43K
 |doi-access=free
 }}
</ref> the results
are &alpha;<sub>1</sub>&nbsp;=&nbsp;&minus;94.82°, &alpha;<sub>2</sub>&nbsp;=&nbsp;&minus;78.29°, and
''s''<sub>12</sub>&nbsp;=&nbsp;18752&nbsp;km.  The midpoint of the geodesic is
&phi;&nbsp;=&nbsp;&minus;7.07°, &lambda;&nbsp;=&nbsp;&minus;159.31°,
&alpha;&nbsp;=&nbsp;&minus;57.45°.

==Gnomonic chart==
[[File:Admiralty Chart No 132 Gnomonic Chart of Indian and Southern Oceans, Published 1914.jpg|thumb|Admiralty Gnomonic Chart of the Indian and Southern Oceans, for use in plotting great circle tracks]]
A straight line drawn on a [[gnomonic projection|gnomonic chart]] is a portion of a great circle. When this is transferred to a [[Mercator projection|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.

==See also==
* [[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]]

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}

==External links==
* [http://mathworld.wolfram.com/GreatCircle.html Great Circle &ndash; 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]]