Great Circles, Rhumb Lines & Directions on the Earth — Page 37, Lesson 43

We’re now going to work through the worked examples that turn the theory of great circles and rhumb lines into actual navigation calculations. These are the classic cases you’ll meet again and again, and each one shows you a different way the angular distance — and therefore the shortest distance — between two points is found. Let’s start with Example 2. We have Durban, position D, at 29°30’ South, 030°30’ East, and Leningrad, position E, at 59°47’ North, 030°30’ East. The first thing you notice is that both longitudes are identical — 030°30’ East — so both places lie on the same meridian. But they are in opposite hemispheres: Durban is south of the Equator, Leningrad is north. When two points share a meridian but sit in different hemispheres, the angular distance between them is simply the sum of the two latitudes. So the change of latitude, which we abbreviate as ch.lat, is 29°30’ plus 59°47’, giving 89°17’ north from D. That’s the angular distance between the two places, measured along the meridian. Now, to turn that angular distance into a linear distance, we use the fact that one minute of latitude equals one nautical mile. So 89 degrees is 89 times 60, which is 5340 minutes, and we add the remaining 17 minutes, giving 5357 nautical miles. That’s the shortest distance between Durban and Leningrad. Now let’s move to Example 3, which introduces a new concept: the anti-meridian. Here we have Rome, position F, at 41°55’ North, 011°10’ East, and Honolulu, position G, at 21°17’ North, 168°50’ West. Both are in the same hemisphere — both north — but they are on meridian and anti-meridian. What does that mean? The two longitude values are of opposite sign — one East, one West — and they add up numerically to 180°. Let’s check: 011°10’ East plus 168°50’ West equals exactly 180°. So Rome and Honolulu are on opposite sides of the globe, on meridians that are 180° apart. Here’s the surprising result: the great circle between Rome and Honolulu passes over the North Pole. That’s a crucial insight — if you wanted to fly from Rome to Honolulu in one stage, the shortest route goes right over the top of the world, not across the Atlantic and Pacific as you might instinctively think. Now, how do we calculate the angular distance in this case? The simplest method is by inspection: add the two latitudes and subtract their total from 180°. So angular distance equals 180° minus (41°55’ plus 21°17’). That’s 180° minus 63°12’, which gives 116°48’. There’s an alternative method, and it’s mathematically the same. We calculate the change of latitude from Rome to the Pole: 90° minus 41°55’ equals 48°05’. Then from the Pole to Honolulu: 90° minus 21°17’ equals 68°43’. Add those two together — 48°05’ plus 68°43’ — and you get the same 116°48’. Both methods give the identical result, because they’re just two ways of looking at the same geometry. Finally, we convert that angular distance to linear distance. 116°48’ is 116 times 60, which is 6960 minutes, plus 48 minutes, giving 7008 nautical miles. And note the direction: the initial direction from Rome to the Pole is north, and then from the Pole to Honolulu it’s south. So the great circle route takes you due north first, over the Pole, then due south. Let me show you the sectional diagram for this example — it makes the geometry much clearer. So the key takeaways from these two examples: when two points share a meridian but are in opposite hemispheres, you add the latitudes. When they’re on meridian and anti-meridian in the same hemisphere, you add the latitudes and subtract from 180°, or equivalently, work out the distance from each point to the Pole and add those. And in every case, one minute of arc equals one nautical mile. That’s the fundamental conversion that ties angular distance to linear distance.