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D1 - - - Great circle sailing by circular approximation - - - 10/03/2004

If a short voyage is to be made (less than 4000 miles) that does not cross the equator, the great-circle can be traveled by means of the approximation formula and the circular approximation.

If the rhumb line heading and the great-circle heading are found, and the smaller value subtracted from the larger value the result is called the great-circle correction angle (Theta sub c) which I will call GCC for great circle correction.

DLO = difference in longitude
Lm  = mean latitude = ((L1+L2)/2)
Ld  = (L2-L1) = latitude difference

    tan GCC = tan (( DLO ) / 2 ) * sin Lm / cos ( Ld / 2 )

A second approximation can be made, giving;

    GCC = ( DLO / 2 ) * sin Lm  (round off to integer value)

If a great-circle is drawn on a Mercator map of the world and it does not lie along either a meridian or the equator it will look like a sign wave circling the world. A short section of that curve that is away from the equator can be considered to approximate a segment of a circle; the rhumb line connecting two points on the circle will be a chord on the circular section; the angle at the intersection of the circle and the chord is the great-circle correction angle (GCC).

If the great circle path is divided into N parts so that we have a sequence of chords or legs to approximate the great-circle then the heading correction ( Hc ) to be applied to the initial rhumb line heading will be as follows.

Hc = ( GCC * (N-1)) / N

Considering that the angles will be small integer values, we can set N = GCC producing the following.

Hc = GCC - 1

Dividing the distance to the final great-circle destination by N we get the distance for each leg.

( The rhumb line distance divided by N can be used as an aproximation)
Dividing the length of each leg by the cruising speed gives the time it should take to travel the leg.

When N is set equal to GCC and there are no errors due to round off or drift, the course change at the end of each leg should be 2 degrees. At the end of each leg the whole computation is repeated to find the new heading.

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