Math CentralQuandaries & Queries


Question from Hervé:

On the earth, the mathematical formula giving the distance between two points, and the initial course for a boat on the great circle is well known.
I need to find the inverse formula, ie knowing an initial position on earth, and the initial course of the boat, and the distance to run on the great circle, the formula gives the final position (longitude and latitude).


You know, as well as your initial longitude, your initial latitude L , your bearing b, and your run r. Your angular distance from the North Pole is thus (90-L), taking South latitudes as negative; this, r, and b give two sides and the included angle of a spherical triangle with vertices at start, finish, and the North Pole.

This may be solved for the remaining side using the spherical cosine law:

\[\cos x = \cos (90-L) \cos r + \sin (90-L) \sin r \cos b\]

with the new latitude L' equal to $90-x$

or (as $\sin(90-a) = \cos(a)$)

\[\sin(L') = \sin L \cos r + \cos L \sin r \cos b\]

Once you have found L', you need your new longitude, which you find from the spherical sine law. If the angle subtended at the pole by the start and finish points is A, then

\[\frac{\sin(A)}{\sin(r)} = \frac{sin(b)}{sin(x)}.\]

The angle A is also the difference between new and old longitudes.

Good Hunting (fishing?)



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