Great-circle navigation

If a ship starts at **t** and sails straight to the North Pole, the travel distance is ::d_{t,N} = R\theta_{t,n} =R(\pi/2-\varphi_t) #### Derivation The *cosine formula* of spherical trigonometry yields for the angle p between the great circles through **s** that point to the North on one hand and to **t** on the other hand ::\cos\theta_{t,N} = \cos\theta_{s,t}\cos\theta_{s,N}+\sin\theta_{s,t}\sin\theta_{s,N}\cos p. ::\sin\varphi_t = \cos\theta_{s,t}\sin\varphi_s +\sin\theta_{s,t}\cos\varphi_s\cos p. The *sine formula* yields ::\frac{\sin p}{\sin \theta_{t,N}} = \frac{\sin(\lambda_t-\lambda_s)}{\sin\theta_{s,t}}. Solving this for sin θs,t and insertion in the previous formula gives an expression for the tangent of the position angle, ::\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; ::\tan p = \frac{\sin(\lambda_t-\lambda_s)\cos\varphi_t\cos\varphi_s}{\sin\varphi_t-\cos\theta_{s,t}\sin\varphi_s}. #### Further details Because the brief derivation gives an angle between 0 and 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 p such that use of the correct branch of the inverse tangent allows to produce an angle in the full range −π ≤ *p* ≤ π. The computation starts from a construction of the great circle between **s** and **t**. It lies in the plane that contains the sphere center, **s** and **t** and is constructed rotating **s** by the angle θs,t around an axis ω. The axis is perpendicular to the plane of the great circle and computed by the normalized vector cross product of the two positions: ::\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). A right-handed tilted coordinate system with the center at the center of the sphere is given by the following three axes: the axis **s**, the axis ::\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) and the axis ω. A position along the great circle is ::\mathbf{s}(\theta) = \cos\theta \mathbf{s}+\sin\theta \mathbf{s}_\perp,\quad 0\le\theta\le 2\pi. The compass direction is given by inserting the two vectors **s** and **s**⊥ and computing the gradient of the vector with respect to θ at . ::\frac{\partial}{\partial\theta}\mathbf{s}_{\mid \theta=0}=\mathbf{s}_\perp. The angle p is given by splitting this direction along two orthogonal directions in the plane tangential to the sphere at the point **s**. The two directions are given by the partial derivatives of **s** with respect to φ and with respect to λ, normalized to unit length: ::\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); ::\mathbf{u}_E = \left(\begin{array}{c} -\sin\lambda_s\\ \cos\lambda_s\\ 0 \end{array} \right); ::\mathbf{u}_N\cdot \mathbf{s} = \mathbf{u}_E\cdot \mathbf{u}_N =0 **u**N points north and **u**E points east at the position **s**. The position angle p projects **s**⊥ into these two directions, ::\mathbf{s}_\perp = \cos p \,\mathbf{u}_N+\sin p\, \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 p are computed by multiplying this equation on both sides with the two unit vectors, ::\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)]; ::\sin p = \mathbf{s}_\perp \cdot \mathbf{u}_E =\frac{1}{\sin\theta_{s,t}}[\cos\varphi_t\sin(\lambda_t-\lambda_s)]. Instead of inserting the convoluted expression of **s**⊥, the evaluation may employ that the triple product is invariant under a circular shift of the arguments: ::\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). If atan2 is used to compute the value, one can reduce both expressions by division through cos φt and multiplication by sin θs,t, because these values are always positive and that operation does not change signs; then effectively ::\tan p = \frac{\sin(\lambda_t-\lambda_s)}{\cos\varphi_s\tan\varphi_t -\sin\varphi_s\cos(\lambda_t-\lambda_s)}. ## Finding way-points To find the way-points, that is the positions of selected points on the great circle between *P*1 and *P*2, 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 :\tan\alpha_0 = \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 ≈ ±π. Let the angular distances along the great circle from *A* to *P*1 and *P*2 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 = 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} ::figure[src="https://upload.wikimedia.org/wikipedia/commons/4/48/Sphere_geodesic_2sigma.svg" caption="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*.{{refn|group=note|The direct geodesic problem, finding the position of *P*2 given *P*1, α1, and *s*12, 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 ≈ ±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 = (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 = +π. It may be convenient to parameterize the route in terms of the longitude using :\tan\phi = \cot\alpha_0\sin(\lambda-\lambda_0).{{refn|group=note| The following is used: \cos\sigma = \cos\phi \cos(\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°,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 *s*12 = 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 = (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, |doi-access=free the results are α1 = −94.82°, α2 = −78.29°, and *s*12 = 18752 km. The midpoint of the geodesic is φ = −7.07°, λ = −159.31°, α = −57.45°. ## Gnomonic chart ::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/78/Admiralty_Chart_No_132_Gnomonic_Chart_of_Indian_and_Southern_Oceans,_Published_1914.jpg" caption="Admiralty Gnomonic Chart of the Indian and Southern Oceans, for use in plotting great circle tracks"] :: 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. ## Notes ## References ## References 1. (7 June 2011). ["Methods and Algorithms in Navigation: Marine Navigation and Safety of Sea Transportation"](https://books.google.com/books?id=buMsPGyE7boC&q=loxodromic+navigation&pg=PA139). *[[CRC Press]]*. 2. {{AS ref. 4.3.149 ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Great-circle_navigation) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Great-circle_navigation?action=history). ::