This article presents what the author believes to be a new method of obtaining the latitudes and longitudes of a series of points on the great-circle track between two given places, by computation.[1]
In the usual method employed for the solution of this problem, it is necessary to determine:
(1) The great-circle distance.
(2) The great-circle initial course.
(3) The latitude and longitude of the vertex (greatest latitude).
(4) The latitude and longitude of a series of points along the great-circle track.
The fourth step is necessary in order to plot the great-circle track, as a series of chords, on a Mercator projection. In fact, this last operation is the main objective of the entire set of calculations.
In the course of the computations described above, the following formulas are employed, in the order indicated, as explained in Dutton’s Navigation and Nautical Astronomy (1939), pages 98-100:
(a) hav d = hav DL0 cos L1 cos L2 + hav (L1 ~L2)
(b) hav C = [hav coL2 —hav (d~co~L1)J csc d sec Li
or sin C = sin DL0 cos L2 csc d
(c) sin co-Lv = sin co-L1 sin C
(d) tan DLv = sec co-L1 cot C
(e) cot Lm = tan co-Lv sec 0
or sec 0 = cot La cot co-Lv
The following standard and additional symbols are used above:
(1) d = distance.
(2) L1 λ= latitude and longitude of point of departure.
(3) L2 λ=latitude and longitude of point of destination.
(4) DL0 = difference of longitude (between point of departure and destination).
(5) C = Initial great circle course angle.
(6) Lv= latitude of the vertex.
(7) DLov = difference of longitude between the vertex and the point of departure.
(8) La = assumed latitude.
(9) 0=the difference of longitude between the vertex and any chosen meridian, on which it is desired to find the latitude (Lm) at which the great-circle track will cross the selected meridian.
The formulas presented below permit of obtaining a series of intersections made by the great-circle track with chosen meridians of longitude (or parallels of latitude) by a more direct process. The solution is based on determining not the vertex, but the longitude of one of the two points at which the great-circle track intersects the equator, which are of course 90° in longitude on either side of the vertex.
In the following formulas some new symbols and their meanings are:
(10) λ=longitude at which great-circle track crosses the equator.
(11) 0 = difference of longitude from λ to any chosen meridian.
(12) DLoe = difference of longitude from λ to λ.
The formulas are:
(f) tan (DLo/2 + λ — λ) = sin (L2+L1) csc (L2—L1) tan DLoe/2.
(g) tan Lm = tan L1 sin 0 esc DLoe or, if a latitude is assumed (La)2 then sin 0 = cot L1 sin DLoe tan La
The derivation of these formulas is as follows—referring to Fig. 1:
[FIG 1.]
[EQUATION]
This formula is a means of finding λ, after solving for (DLo/2+ λ — λ). Now, if a longitude be assumed, and the difference of longitude from λ to the assumed longitude be called d, then,
tan Lm=tan L1 sin 0 csc DLoe
or if latitudes are assumed, then
sin 0 = tan La cot L1 sin DLoe
When the great-circle track runs generally east and west, it is better to assume longitudes, and when it runs more nearly north and south, to assume latitudes.
In using these formulas, strict attention must be paid to the signs of the angles and of the various functions. In the foregoing, north latitudes and east longitudes are considered positive (+), and south latitudes and west longitudes negative (—). An easterly difference of longitude is (+) and a westerly one (—).
To illustrate and compare the two methods of solution, a typical problem is shown worked out completely by each process.
Problem: It is required to determine the latitudes at which the great-circle track
[EQUATION]
will intersect the meridians of 153°, 143°, 138°, 133°, and 128° W.
[EQUATIOIN]
For all practical purposes the latitudes can be rounded off to the nearest tenth of a degree.
[EQUATION]
It is evident that the latter method is a shorter one, when only a series of points on the great-circle track is desired. The foregoing example was worked as accurately as five-place tables permit, to demonstrate that the results are to all purposes identical. After plotting this series of points on the Mercator projection, the various courses and distances between the points are obtained from the Mercator chart or plotting sheet.
The following examples are given to illustrate various cases of positive and negative values of latitudes, longitudes, and differences of longitude.
[TABLE]
[1] The author wishes to acknowledge the generous and valuable assistance of Lieutenant Delwyn Hyatt, U. S. Navy (Retired), of the Department of Seamanship and Navigation, U. S. Naval Academy, in preparing this article for publication.