Great-circle navigation

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 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 cosine of the angle is calculated by the dot product of the two vectors If the ship steers straight to the North Pole, the travel distance is If a ship starts at t and swims straight to the North Pole, the travel distance is The cosine formula of spherical trigonometry[4] 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 The sine formula yields Solving this for sin θs,t and insertion in the previous formula gives an expression for the tangent of the position angle, 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 values of the cosine and sine of p are computed by multiplying this equation on both sides with the two unit vectors, 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: 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 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 longitude at the node is found from Finally, calculate the position and azimuth at an arbitrary point, P (see Fig.

Likewise, the vertex, the point on the great circle with greatest latitude, is found by substituting σ = +1⁄2π.

The path determined in this way gives the great ellipse joining the end points, provided the coordinates

The positions are transferred at a convenient interval of longitude and this track is plotted on the Mercator chart for navigation.

Orthodromic course drawn on the Earth globe
Figure 1. The great circle path between (φ 1 , λ 1 ) and (φ 2 , λ 2 ).
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 u E , u N and the rotation axis ω are marked by arrows.
Figure 2. The great circle path between a node (an equator crossing) and an arbitrary point (φ,λ).
Admiralty Gnomonic Chart of the Indian and Southern Oceans, for use in plotting great circle tracks