I’ve just submitted a new article for publication in the boating press and I’ve decided to share a longer, more detailed version of it with you. You may find it useful to calculate Great Circle distances between widely separated points on the planet, and this will give you a way to do it without having direct access to the Internet (like a yachtsman out at sea). The article, if published, will lack the example calculation I have given here, but you may find it helpful. All you need is the lat and long of your starting point and destination, and a pocket calculator.
Great Circle Navigation by Pocket Calculator
by Elisee Reclus
If you want to fly an aircraft from Mexico City to Moscow, common sense tells that you are going to have to fly a Great Circle (GC) over the Arctic Ocean. But if you plot the same voyage on a standard Mercator nautical chart, the straight line (or rhumb line) goes nowhere near the pole. The Mercator map projection distorts the shape of the Earth in order to make the rhumb lines consistent with compass courses. For short distances, the rhumb line is very close to the Great Circle, but over long distances, steering the Great Circle may result in substantial savings in time, distance and fuel. On the surface of a sphere, the GC is the shortest distance between two points.
When you ask your onboard navigational computer for the course and distance from point A to point B, the distance will be in nautical miles, and the course to be steered will be referenced to the Great Circle route. For shorter voyages, the difference between rhumb line and GC is negligible. But on longer ocean passages, it may be preferable to steer along the GC course. The disadvantage is that the GC course is constantly changing relative to the parallels and meridians on the chart. The navigator has to change his ship’s heading frequently to stay near the GC path.
Determining the GC course requires using specialized map projections or complex mathematical calculations, but there exists a relatively simple method which can be employed, requiring only a pocket calculator. The calculator must be able to handle simple trigonometric calculations; sine, cosine, and inverse functions. This type of simple student calculator is cheap (Casio makes a solar-powered model for about $12) and is available anywhere in the world.
The GC calculation is carried out using the same sight reduction algorithm used to determine the azimuth and intercept of a sextant sight. Incidentally, I highly recommend this method as a backup to your GPS and sight reduction tables. The calculations are explained in detail in Sight Reduction Procedures, 6. The Calculated Altitude and Azimuth, page 279 of the Nautical Almanac. Those of you who reduce your sextant sights with a pocket calculator are no doubt already familiar with this method. The GC algorithm requires as input the Greenwich Hour Angle (GHA) and Declination (Dec) of the body, and the Assumed Latitude and Longitude (ALat and ALon) of your Assumed Position (AP). It is not necessary to fudge the AP to meet the requirements of your sight reduction tables, the calculator algorithm will accept any AP within a 100 miles or so of your true (but unknown) position. You can use your DR position, or any convenient round number, even a lat/lon grid intersection on your chart. The algorithm returns the Calculated Azimuth (Az) and Calculated Altitude (Hc) for that body.
To use this algorithm for GC work, simply replace GHA with the longitude of your destination and Dec with the latitude of your destination. (Positive for N, negative for S of the equator). Keep in mind that GHA is always positive, measured to the west, so if the longitude of your destination is east of Greenwich, it must be subtracted from 360 degrees. So for example, if the longitude of your destination is 15 degrees east, its “GHA” is 345 degrees. For west longitudes of the destination, the GHA and longitude are the same.
The latitude and longitude of your point of departure are entered in ALat and ALon respectively, with the appropriate sign for N/S,E/W. The algorithm will return the Az (the compass course to be steered) and Hc. Subtract Hc from 90 degrees and convert to minutes of arc and this will be the GC distance from departure to destination in nautical miles.
Keep in mind that steering a Great Circle means your course will constantly change, so every time you calculate a fix en route you need to update this calculation and derive a new direction and distance to your destination.
Example: New York (40d 49′N, 73d 58′W) to Paris (48d 50′N, 2d 20′E)
So the GHA of Paris is 357d 40′
Solution (Using the sight reduction algorithm described in page 279 of the Nautical Almanac).
Ho = 90d
Dec = 48d 50′ , GHA = 357d 40′
ALat = 40d 49′, ALon = -73d 58′
LHA = GHA + ALon = 283d 42′, cos LHA = 0.23683
Cos Dec = 0.65825, cos Dec x cos LHA = 0.15589 = C
Cos ALat = 0.75680, sin ALat = 0.65364, sin Dec = 0.75280 = S
Intercept Calculation (Great Circle Distance from departure to destination)
Hc = INVsin ( (S x sin ALat) + (C x cos ALat) )
Hc = 37d 35′
Cos Hc = 0.79251
Ho – Hc = 52d 25′ = 3145 nautical miles
Azimuth Calculation (Initial course from departure to destination)
X = ( (S x cos ALat) – (C x sin Alat) ) / cos Hc
X = 0.59031/0.79251
A = INVcos X = 054d
If LHA > 180d then Az = A
Otherwise, Az = 360d – A
Your results may differ slightly due to round-off errors.