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.

APPROXIMATION FORMULAS

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)

GREAT CIRCLE SAILING BY CIRCULAR APPROXIMATION

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.