Great Circles, Rhumb Lines & Directions on the Earth — Page 37, Lesson 45

I want to walk you through two worked examples that show how we calculate the shortest distance between two points on the Earth's surface. These build directly on the great-circle principles we've been using. Let's start with Example 4, which deals with a meridian and anti-meridian case where the two points are in different hemispheres. Our two cities are Tokyo, at 35 degrees 57 minutes north, 135 degrees 35 minutes east, and Rio de Janeiro, at 22 degrees 10 minutes south, 044 degrees 25 minutes west. First, notice that the longitudes have opposite signs — one east, one west — and they add up to 180 degrees. That tells us this is another meridian and anti-meridian situation, but this time the latitudes are in different hemispheres: one north, one south. The key step is to draw a sectional diagram — you can see this in Figure 2.12. If you sketch the angles reasonably accurately, you can see by inspection that the shortest distance between Tokyo and Rio goes via the North Pole. So we base our calculation on that route. Let me walk through the angular distances. From Tokyo to the North Pole, we travel north through an angular distance of 90 degrees minus 35 degrees 57 minutes, which gives us 54 degrees 03 minutes north. From the North Pole down to the Equator is a further 90 degrees, but now we're travelling south. Then from the Equator down to Rio de Janeiro is 22 degrees 10 minutes south. Adding these three stages together gives a total angular distance of 166 degrees 13 minutes. To convert that angular distance into a linear distance in nautical miles, remember that one minute of arc along a great circle equals one nautical mile. So we take 166 degrees times 60 minutes per degree, which gives us 9,960 minutes, then add the remaining 13 minutes, for a total of 9,973 nautical miles. Now, what if we had initially decided to route via the South Pole instead? Let's check that. From Tokyo to the South Pole would be 35 degrees 57 minutes south, then another 90 degrees south to the South Pole, then from the South Pole up to Rio would be 90 degrees minus 22 degrees 10 minutes, which is 67 degrees 50 minutes north. Adding those gives 193 degrees 47 minutes. Since that angular distance is greater than 180 degrees, this is the longer way around the Earth. Here's a useful trick: if you accidentally calculate the longer angular distance, you don't need to redo the whole calculation. Simply subtract the longer angular distance from 360 degrees to get the correct answer. So 360 degrees minus 193 degrees 47 minutes equals 166 degrees 13 minutes — the same answer we got before. Alternatively, you can continue the wrong calculation to get a linear distance: 193 degrees times 60 plus 47 minutes gives 11,627 nautical miles. Then subtract that from the circumference of the Earth, which is 21,600 nautical miles. So 21,600 minus 11,627 gives you 9,973 nautical miles — the same correct answer. And remember to also change the directions involved when you switch your route. Let's move to Example 5, which is simpler — two points on the Equator. Our cities are Dakar at 0 degrees 0 minutes north, 016 degrees 35 minutes west, and Singapore at 0 degrees 0 minutes north, 103 degrees 55 minutes east. This example uses the fact that a minute of longitude equals one nautical mile, but only at the Equator. The sectional diagram is drawn in the plane of the Equator, viewed from above the North Pole, and for reference we draw the Prime Meridian at 0 degrees east/west and the Anti-Prime Meridian at 180 degrees east/west — you can see this in Figure 2.13. The change of longitude from Dakar to Singapore is 016 degrees 35 minutes in an easterly direction to the Prime Meridian, then a further 103 degrees 55 minutes in an easterly direction to Singapore. That gives a total easterly change of longitude of 120 degrees 30 minutes. At the Equator only, that angular measurement of 120 degrees 30 minutes converts directly to a linear measurement: 120 degrees times 60 plus 30 minutes gives 7,230 nautical miles. That's the shortest distance between Dakar and Singapore, because they're both on the Equator and the great-circle route follows the Equator itself.