Great-circle navigation

For the navigation on an ellipsoid, see Geodesics on an ellipsoid.

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 often accurate enough for navigation (see the numerical example).

Course and distance

Figure 1. The great circle path between (φ₁, λ₁) and (φ₂, λ₂).

The great circle path may be found using spherical trigonometry; this is the spherical version of the inverse geodesic 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{aligned}\tan \alpha _{1}&={\frac {\sin \lambda _{12}}{\cos \phi _{1}\tan \phi _{2}-\sin \phi _{1}\cos \lambda _{12}}},\\\tan \alpha _{2}&={\frac {\sin \lambda _{12}}{-\cos \phi _{2}\tan \phi _{1}+\sin \phi _{2}\cos \lambda _{12}}},\\\end{aligned}}

where λ₁₂ = λ₂ − λ₁^{[note 1]} 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

\cos \sigma _{12}=\sin \phi _{1}\sin \phi _{2}+\cos \phi _{1}\cos \phi _{2}\cos \lambda _{12}.

^{[note 2]}^{[note 3]}

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, yields results for the distance s₁₂ which are within 1% of the geodesic distance for the WGS84 ellipsoid.

Finding way-points

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 the spherical sine rule:

\sin \alpha _{0}=\sin \alpha _{1}\cos \phi _{1}.

^{[note 4]}

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 = ½π, use σ₀₁ = 0).

This gives σ₀₁, whence σ₀₂ = σ₀₁ + σ₁₂.

The longitude at the node is found from

{\begin{aligned}\tan \lambda _{01}&={\frac {\sin \alpha _{0}\sin \sigma _{01}}{\cos \sigma _{01}}},\\\lambda _{0}&=\lambda _{1}-\lambda _{01}.\end{aligned}}

Figure 2. The great circle path between a node (an equator crossing) and an arbitrary point (φ,λ).

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

{\color {white}.\,\qquad )}\sin \phi =\cos \alpha _{0}\sin \sigma ,

^{[note 6]}

{\begin{aligned}\tan(\lambda -\lambda _{0})&={\frac {\sin \alpha _{0}\sin \sigma }{\cos \sigma }},\\\tan \alpha &={\frac {\tan \alpha _{0}}{\cos \sigma }}.\end{aligned}}

The atan2 function should be used to determine σ₀₁, λ, and α. For example, to find the midpoint of the path, substitute σ = ½(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 = +½π. It may be convenient to parameterize the route in terms of the longitude using

\tan \phi =\cot \alpha _{0}\sin(\lambda -\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 $(\phi ,\lambda )$ 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

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°, 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°, σ₁ = −96.76°, σ₂ = 71.8°, λ₀₁ = 98.07°, and λ₀ = −169.67°. Then to compute the midpoint of the route (for example), take σ = ½(σ₁ + σ₂) = −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,^[3] 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°.

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.

Notes

↑ In 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.
↑ If σ₁₂ is less than 0.01° (or more than 179.99°) its cosine is 0.99999999 or more, leading to some inaccuracy. To avoid this, replace the usual formula with
$\tan \sigma _{12}={\frac {\sqrt {(\cos \phi _{1}\sin \phi _{2}-\sin \phi _{1}\cos \phi _{2}\cos \lambda _{12})^{2}+(\cos \phi _{2}\sin \lambda _{12})^{2}}}{\sin \phi _{1}\sin \phi _{2}+\cos \phi _{1}\cos \phi _{2}\cos \lambda _{12}}}.$
The numerator of this formula contains the quantities that were used to determine tanα₁.
↑ These equations for α₁,α₂,σ₁₂ are suitable for implementation on modern calculators and computers. For hand computations with logarithms, Delambre's analogies^[1] were usually used:
${\begin{aligned}\cos {\tfrac {1}{2}}(\alpha _{2}+\alpha _{1})\sin {\tfrac {1}{2}}\sigma _{12}&=\sin {\tfrac {1}{2}}(\phi _{2}-\phi _{1})\cos {\tfrac {1}{2}}\lambda _{12},\\\sin {\tfrac {1}{2}}(\alpha _{2}+\alpha _{1})\sin {\tfrac {1}{2}}\sigma _{12}&=\cos {\tfrac {1}{2}}(\phi _{2}+\phi _{1})\sin {\tfrac {1}{2}}\lambda _{12},\\\cos {\tfrac {1}{2}}(\alpha _{2}-\alpha _{1})\cos {\tfrac {1}{2}}\sigma _{12}&=\cos {\tfrac {1}{2}}(\phi _{2}-\phi _{1})\cos {\tfrac {1}{2}}\lambda _{12},\\\sin {\tfrac {1}{2}}(\alpha _{2}-\alpha _{1})\cos {\tfrac {1}{2}}\sigma _{12}&=\sin {\tfrac {1}{2}}(\phi _{2}+\phi _{1})\sin {\tfrac {1}{2}}\lambda _{12}.\end{aligned}}$
McCaw^[2] 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).
↑ To obtain accurate results in all cases, particularly when α₀ ≈ ±½π, use instead
$\tan \alpha _{0}={\frac {\sin \alpha _{1}\cos \phi _{1}}{\sqrt {\cos ^{2}\alpha _{1}+\sin ^{2}\alpha _{1}\sin ^{2}\phi _{1}}}}.$
↑ 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,
${\begin{aligned}\sigma _{12}&=s_{12}/R,\\\sin \phi _{2}&=\sin \phi _{1}\cos \sigma _{12}+\cos \phi _{1}\sin \sigma _{12}\cos \alpha _{1},\quad {\text{or}}\\\tan \phi _{2}&={\frac {\sin \phi _{1}\cos \sigma _{12}+\cos \phi _{1}\sin \sigma _{12}\cos \alpha _{1}}{\sqrt {(\cos \phi _{1}\cos \sigma _{12}-\sin \phi _{1}\sin \sigma _{12}\cos \alpha _{1})^{2}+(\sin \sigma _{12}\sin \alpha _{1})^{2}}}},\\\tan \lambda _{12}&={\frac {\sin \sigma _{12}\sin \alpha _{1}}{\cos \phi _{1}\cos \sigma _{12}-\sin \phi _{1}\sin \sigma _{12}\cos \alpha _{1}}},\\\lambda _{2}&=\lambda _{1}+\lambda _{12},\\\tan \alpha _{2}&={\frac {\sin \alpha _{1}}{\cos \sigma _{12}\cos \alpha _{1}-\tan \phi _{1}\sin \sigma _{12}}}.\end{aligned}}$
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.
↑ To obtain accurate results in all cases, particularly when φ ≈ ±½π, use instead
$\tan \phi ={\frac {\cos \alpha _{0}\sin \sigma }{\sqrt {\cos ^{2}\sigma +\sin ^{2}\alpha _{0}\sin ^{2}\sigma }}}.$

References

↑ 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". J. Geodesy. 87 (1): 43–55. doi:10.1007/s00190-012-0578-z. Retrieved 2012-07-14.

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.

This article is issued from Wikipedia - version of the 10/12/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.