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Great-circle navigation is the practice of navigating a vessel (a ship or aircraft) along a great circle. A great circle track is the shortest distance between two points on the surface of a sphere; the Earth isn't exactly spherical, but the formulas for a sphere are simpler and are accurate enough for navigation.

Methods

To calculate a great circle track the navigator may use several methods.

Computer software

A navigator can specify departure and arrival positions and software will create a list of waypoints on the great circle track. Such programs calculate the total distance, the distance between successive waypoints, and the courses between successive waypoints.

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.

Course and distance

The great circle path may be found using spherical trigonometry. If a navigator begins at $(\phi _{1},\lambda _{1})$ and plans to travel the great circle to a point at point $(\phi _{2},\lambda _{2})$ ( $\phi$ is the latitude and $\lambda$ is the longitude), the initial and final courses $\alpha _{1}$ and $\alpha _{2}$ are given by^[1]

{\begin{aligned}x_{1}&=\cos \phi _{1}\sin \phi _{2}-\sin \phi _{1}\cos \phi _{2}\cos \lambda _{12},\\y_{1}&=\cos \phi _{2}\sin \lambda _{12},\\x_{2}&=-\sin \phi _{1}\cos \phi _{2}+\cos \phi _{1}\sin \phi _{2}\cos \lambda _{12},\\y_{2}&=\cos \phi _{1}\sin \lambda _{12},\\\tan \alpha _{1}&={\frac {y_{1}}{x_{1}}},\\\tan \alpha _{2}&={\frac {y_{2}}{x_{2}}},\\\end{aligned}}

where $\lambda _{12}=\lambda _{2}-\lambda _{1}$ and the quadrants of $\alpha _{1},\alpha _{2}$ 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, $\sigma _{12}$ , is given by

\tan \sigma _{12}={\frac {\sqrt {x_{1}^{2}+y_{1}^{2}}}{\sin \phi _{1}\sin \phi _{2}+\cos \phi _{1}\cos \phi _{2}\cos \lambda _{12}}}.

The distance along the great circle will then be $R\sigma _{12}$ , where $R$ is the assumed mean earth radius and $\sigma _{12}$ is expressed in radians.

Navigating the route

Having found $\alpha _{1}$ , the full route may be easily determined. Let the node, the point at which the great circle crosses the equator in the northward direction, be $(\phi =0,\lambda _{0})$ . The azimuth at this point, $\alpha _{0}$ , is given by

\tan \alpha _{0}={\frac {\sin \alpha _{1}\cos \phi _{1}}{\sqrt {\cos ^{2}\alpha _{1}+\sin ^{2}\alpha _{1}\sin ^{2}\phi _{1}}}},

which is a variation of Clairaut's relation which allows nearly equatorial great circles to be treated accurately. Parametrize points along the great circle in terms of $\sigma _{p}$ , the arc distance along the great circle from the node to an arbitrary point $p$ . $\sigma _{1}$ and $\sigma _{2}$ are given by

{\begin{aligned}\tan \sigma _{1}&={\frac {\sin \phi _{1}}{\cos \alpha _{1}\cos \phi _{1}}},\\\sigma _{2}&=\sigma _{1}+\sigma _{12}.\end{aligned}}

Finally, the position and azimuth at point $p$ are given by^[2]

{\begin{aligned}\tan \beta _{p}&={\frac {\cos \alpha _{0}\sin \sigma _{p}}{\sqrt {\cos ^{2}\alpha _{0}\cos ^{2}\sigma _{p}+\sin ^{2}\alpha _{0}}}},\\\tan \lambda _{0p}&={\frac {\sin \alpha _{0}\sin \sigma _{p}}{\cos \sigma _{p}}},\\\tan \alpha _{p}&={\frac {\sin \alpha _{0}}{\cos \alpha _{0}\cos \sigma _{p}}},\end{aligned}}

where $\lambda _{0p}=\lambda _{p}-\lambda _{0}$ . The atan2 function should be used to determine $\sigma _{1}$ , $\lambda _{0p}$ , and $\alpha _{p}$ . The longitude of the node is given by $\lambda _{0}=\lambda _{1}-\lambda _{01}$ . The vertex, the point on the great circle with greatest latitude, is found by substituting $\sigma _{p}=\sigma _{v}=+{\frac {1}{2}}\pi$ .

These formulas apply to a spherical model of the earth. On an ellipsoid of revolution, the shortest path is a geodesic and this is normally found by transferring the geodesic on the ellipsoid to a great circle on an auxiliary sphere.^[3]

References

^ Vincenty, T. (1975a). "Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations" (PDF). Survey Review. XXIII (misprinted as XXII) (176): 88–93. Retrieved 2009-07-11. {{cite journal}}: Unknown parameter |month= ignored (help)
^ Karney, C. F. F. (2013). "Algorithms for geodesics". J. Geodesy. 87 (1): 43–55. doi:10.1007/s00190-012-0578-z. Retrieved 2012-07-14.
^ Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)". Astron. Nachr. 331 (8): 852–861. arXiv:0908.1824. doi:10.1002/asna.201011352. English translation of Astron. Nachr. 4, 241–254 (1825).{{cite journal}}: CS1 maint: postscript (link)

Resources

The Great Circle Mapper Displays Great Circle flight routes on a Map And calculates distance and duration
Great Circle – from MathWorld Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
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.

[1] Vincenty, T. (1975a). "Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations" (PDF). Survey Review. XXIII (misprinted as XXII) (176): 88–93. Retrieved 2009-07-11. {{cite journal}}: Unknown parameter |month= ignored (help)

[2] Karney, C. F. F. (2013). "Algorithms for geodesics". J. Geodesy. 87 (1): 43–55. doi:10.1007/s00190-012-0578-z. Retrieved 2012-07-14.

[3] Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)". Astron. Nachr. 331 (8): 852–861. arXiv:0908.1824. doi:10.1002/asna.201011352. English translation of Astron. Nachr. 4, 241–254 (1825).{{cite journal}}: CS1 maint: postscript (link)

[1]

[2]

[3]

@@ Line 10: / Line 10: @@
 A straight line drawn on a [[Map projection#Gnomonic|Gnomonic chart]] would be a great circle track. When this is transferred to a [[Map projection#Mercator|Mercator]] chart, it becomes a curve. The positions are transferred at a convenient interval of [[longitude]] and this is plotted on the Mercator chart.
-===Spherical trigonometry===
+===Course and distance===
-The above methods are based on [[spherical trigonometry]]. If a navigator begins at latitude <math>\scriptstyle\phi_s\,\!</math> and plans to travel the great circle to a point at latitude <math>\scriptstyle\phi_f\,\!</math> with a longitude difference between the points of <math>\scriptstyle\Delta\lambda\,\!</math> (positive eastward), his initial course <math>\alpha_s\,\!</math> is given by
+The great circle path may be found using [[spherical trigonometry]]. If a navigator begins at <math>(\phi_1,\lambda_1)</math> and plans to travel the great circle to a point at point <math>(\phi_2,\lambda_2)</math> (<math>\phi</math> is the latitude and <math>\lambda</math> is the longitude), the initial and final courses <math>\alpha_1</math> and <math>\alpha_2</math> are given by<ref>
-:<math>\begin{align}S\!A&=\cos(\phi_f)\sin(\Delta\lambda);\\
+{{cite journal
-S\!B&=\cos(\phi_s)\sin(\phi_f)-\sin(\phi_s)\cos(\phi_f)\cos(\Delta\lambda);{}_{\color{white}.}\\
+ |first=T. |last=Vincenty |authorlink=Thaddeus Vincenty
-\tan(\alpha_s)&=\frac{S\!A}{S\!B};{}_{\color{white}.}\end{align}\,\!</math>
+ |title=Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations
+ |journal=Survey Review
+ |volume=XXIII (misprinted as XXII) |issue=176 |month=April |year=1975a |pages=88&ndash;93
+ |url=http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf |accessdate=2009-07-11
+}}</ref>
+:<math>\begin{align}
-The [[central angle]] between the two points, <math>\Delta\sigma</math>, is given by
+x_1&=\cos\phi_1\sin\phi_2-\sin\phi_1\cos\phi_2\cos\lambda_{12},\\
-:<math>\tan(\Delta\sigma)=\tan(\sigma_f-\sigma_s)=\frac{\sqrt{S\!A^2+S\!B^2}}{\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)};{}_{\color{white}.}\,\!</math>
+y_1&=\cos\phi_2\sin\lambda_{12},\\
+x_2&=-\sin\phi_1\cos\phi_2+\cos\phi_1\sin\phi_2\cos\lambda_{12},\\
+y_2&=\cos\phi_1\sin\lambda_{12},\\
+\tan\alpha_1&=\frac{y_1}{x_1},\\
+\tan\alpha_2&=\frac{y_2}{x_2},\\
+\end{align}</math>
+where <math>\lambda_{12} = \lambda_2 - \lambda_1</math> and the quadrants of <math>\alpha_1, \alpha_2</math> are determined by the signs of the numerator and denominator in the tangent formulas (e.g., using the [[atan2]] function).
-which involves the [[spherical law of cosines]]. The distance along the great circle will then be <math>\Delta\sigma\,\!</math> times the assumed earth radius, where <math>\Delta\sigma\,\!</math> is in radians—that is, degrees multiplied by <math>\tfrac{\pi}{180}\,\!</math>.
+The [[central angle]] between the two points, <math>\sigma_{12}</math>, is given by
+:<math>\tan\sigma_{12}=\frac{\sqrt{x_1^2 + y_1^2}}{\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\lambda_{12}}.</math>
+The distance along the great circle will then be <math>R\sigma_{12}</math>, where <math>R</math> is the assumed [[Earth radius#Mean radius|mean earth radius]] and <math>\sigma_{12}</math> is expressed in [[Radian#Conversions|radians]].
-===Spherical Trig (the simple version)===
-A navigator starting at latitude <math>\scriptstyle\phi_1\,\!</math> plans to travel the great circle to a point at latitude <math>\scriptstyle\phi_2\,\!</math>, with a longitude difference between the points of L (positive eastward). His initial course <math>\alpha\,\!</math> is given by
+===Navigating the route===
-:<math>\tan \alpha
-= \frac{\sin L}{(\cos \phi_1)(\tan \phi_2)- (\sin\phi_1)(\cos L)}</math><br><br>
-The [[central angle]] <math> \sigma </math> between the two points is given by
+Having found <math>\alpha_1</math>, the full route may be easily determined.  Let the ''node'', the point
+at which the great circle crosses the
+equator in the northward direction, be <math>(\phi=0, \lambda_0)</math>.  The azimuth at this point, <math>\alpha_0</math>, is
+given by
+:<math> \tan\alpha_0 = \frac
-:<math>\cos \;\sigma = (\cos \phi_1)(\cos \phi_2)(\cos L) + (\sin \phi_1)(\sin \phi_2)</math>
+{\sin\alpha_1 \cos\phi_1}{\sqrt{\cos^2\alpha_1 + \sin^2\alpha_1\sin^2\phi_1}},
+</math>
+which is a variation of [[Clairaut's relation]] which allows nearly equatorial great circles to be treated accurately.
-The distance along the great circle will then be <math> \sigma </math> times the assumed earth radius, where <math> \sigma </math> is in radians—that is, degrees multiplied by <math> \pi / 180 </math>.
+Parametrize points along the great circle in terms of <math>\sigma_p</math>, the arc distance along the great circle from the node
+to an arbitrary point <math>p</math>.  <math>\sigma_1</math> and <math>\sigma_2</math> are
+given by
+:<math>
-The earth's actual radius of curvature varies by 1%, so this calculated distance might well be off by a few tenths of a percent; if that's not good enough the navigator can use the ellipsoid-surface formulas in the [[geographical distance]] article.
+\begin{align}
+\tan\sigma_1 &= \frac{\sin\phi_1}{\cos\alpha_1\cos\phi_1},\\
+\sigma_2 &= \sigma_1 + \sigma_{12}.
+\end{align}
+</math>
+Finally, the position and azimuth at point <math>p</math> are given by<ref>
-If the central angle is very close to zero or 180 degrees—if the origin and destination are, say, a kilometer apart, or 19999 kilometers apart—then the cosine of the central angle will be 0.99999999 or thereabouts, leading to some inaccuracy. The more complicated formulas above are intended to cover that situation and are otherwise unnecessary.
+{{cite journal
+ |first=C. F. F.
+ |last=Karney
+ |title=Algorithms for geodesics
+ |journal=J. Geodesy
+ |volume = 87
+ |number = 1
+ |year = 2013
+ |pages = 43&ndash;55
+ |accessdate=2012-07-14
+ |doi=10.1007/s00190-012-0578-z
+ |url=http://dx.doi.org/10.1007/s00190-012-0578-z
+}}
+</ref>
+:<math>\begin{align}
-To find the lat-lons of points along the great circle, start by finding the latitude and longitude of the vertex, the point where the great circle is farthest from the equator:
+\tan\beta_p &= \frac
+{\cos\alpha_0\sin\sigma_p}{\sqrt{\cos^2\alpha_0\cos^2\sigma_p + \sin^2\alpha_0}},\\
+\tan\lambda_{0p} &= \frac
+{\sin\alpha_0\sin\sigma_p}{\cos\sigma_p},\\
+\tan\alpha_p &= \frac
+{\sin\alpha_0}{\cos\alpha_0\cos\sigma_p},
+\end{align}
+</math>
+where <math>\lambda_{0p} = \lambda_p - \lambda_0</math>. The [[atan2]] function should be used to determine
-:<math> \cos \phi_V = (\cos \phi_1)(\sin \alpha) \,</math><br><br>
-:<math> \sin L_V = \frac{\cos \alpha}{\sin \phi_V}</math>
+<math>\sigma_1</math>,
+<math>\lambda_{0p}</math>, and <math>\alpha_p</math>.
+The longitude of the node is given by <math>\lambda_0 = \lambda_1 - \lambda_{01}</math>.  The ''vertex'', the point on the great
+circle with greatest latitude, is found by substituting <math>\sigma_p = \sigma_v = +\frac12\pi</math>.
+These formulas apply to a spherical model of the earth.  On an ellipsoid of revolution, the shortest
-where <math>L_V</math> is the difference in longitude between the navigator's starting point and the vertex. Then
+path is a geodesic and this is normally found by transferring the geodesic on the ellipsoid
+to a great circle on an ''auxiliary sphere''.<ref>
-:<math> \cos X = \frac {\tan \phi_X}{\tan \phi_V}</math>
+{{cite journal
+ |first=F. W. |last=Bessel |authorlink=Friedrich Bessel
-The great circle crosses latitude <math>\phi_X</math> at longitude X east or west of the vertex. For example, if the vertex is at latitude 45 deg then the great circle crosses latitude 44 degrees at longitudes 15.05 deg east and west of the vertex. (All these formulas assume a spherical earth, of course; no chance formulas for the spheroid would be this simple.)
+ |title=The calculation of longitude and latitude from geodesic measurements (1825)
+ |journal=Astron. Nachr.
-===The great circle's node and vertex===
+ |year=2010
-Of particular interest is the great circle's '''''node''''', which is the point of the circle that crosses the equator, and its '''''vertex''''', <math>\phi_v\,\!</math>, which is 90° away from the node and is the point of the circle that is closest to the pole:<ref>[http://www.nordian.no/pdf/easa_general_navigation_demo.pdf The Rhumb Line and the Great Circle in Navigation] pp.4.1-4.3</ref>  If the standpoint is on the equator——i.e., at the great circle's node, <math>\phi_s=0\,\!</math>——and <math>\Delta\lambda\,\!</math> is 90°, then <math>\phi_v=\phi_f\,\!</math>.  The complement of <math>\phi_v\,\!</math> is the azimuth of the great circle at its node, or '''''arc path''''', <math>\alpha_0\,\!</math> or <math>A\,\!</math>:
+ |volume=331 |issue=8 |pages=852&ndash;861
-:<math>\sin(A)=\cos(\phi_v)=\cos(\phi_s)\sin(\alpha_s)=\cos(\phi_f)\sin(\alpha_f)=\cos(\phi_p)\sin(\alpha_p);\,\!</math>
+ |doi=10.1002/asna.201011352
-:<small>(i.e., the latitude cosine of any point on the great circle times the sine of the corresponding azimuth at that point)</small>
+ |arxiv=0908.1824 |postscript=.  English translation of Astron. Nachr. '''4''', 241&ndash;254 (1825).
-Or,
+}}</ref>
-:<math>\begin{matrix}A&=&90^\circ-\phi_v&=&\arctan\Big(\frac{cos(\phi_s)SA}{\sqrt{(sin(\phi_s)SA)^2+SB^2}}\Big);\\
-\phi_v&=&90^\circ-A&=&\arctan\Big(\frac{\sqrt{(sin(\phi_s)SA)^2+SB^2}}{cos(\phi_s)SA}\Big);\end{matrix}\,\!</math>
-This element is the basis of [[Clairaut's relation]]<ref>[http://old.gps.aau.dk/downloads/notes.pdf Ellipsoidal Geometry and Conformal Mapping] pp.13-15 (eq.1.65)</ref>
-While <math>\phi_v\,\!</math> is the vertex latitude, <math>\Delta\lambda_{sv}\,\!</math> is the longitude difference between <math>\lambda_s\,\!</math> and the vertex longitude, <math>\lambda_v\,\!</math>:
-:<math>\sin(\Delta\lambda_{sv})=\frac{\cos(\alpha_s)}{\cos(A)}=\frac{\cos(\alpha_s)}{\sin(\phi_V)}\,\!</math>
-Then
-:<math>\cos(\lambda_p)=\tan(A)\tan(\phi_p)=\tan(\phi_p)\cot(\phi_V)\,\!</math>
-Thus the great circle crosses latitude <math>\phi_p\,\!</math> at <math>\lambda_p\,\!</math> east or west of the vertex. For example, if the vertex is at latitude 45° then the great circle crosses latitude 44° at longitudes 15.05° east and west of the vertex.
-Just as there are different "flavors" of latitude——<math>\phi\,\!</math> being the geographic or geodetic——there are also different flavors of <math>\sigma\,\!</math> and <math>A\,\!</math>.  As such, <math>\widehat{\sigma}\,\!</math> and <math>\widehat{A}\,\!</math> can be used to denote the great circle/spherical valuations of <math>\sigma\,\!</math> and <math>A\,\!</math>.
 ==See also==
@@ Line 70: / Line 108: @@
 * [[Great circle distance]]
 * [[Rhumb line]]
+* [[Geographical distance]]
 ==References==