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

Right, let's pick this up with the two worked examples that close out this section on great-circle distances. We've already dealt with the cases where the two points are on the same meridian, and where they're on meridians that are 180° apart but in the same hemisphere. Now I want to walk you through the genuinely tricky one — Example 4 — which is the meridian and anti-meridian case where the latitudes are in different hemispheres. The problem is this: find the shortest distance between Tokyo, at 35°57' North, 135°35' East, and Rio de Janeiro, at 22°10' South, 044°25' West. The first thing to notice is that the longitudes have opposite signs — one East, one West — and they add up to 180°. 135°35' plus 44°25' is exactly 180°. So this is another meridian and anti-meridian case, but now the latitudes are in different hemispheres, which is what makes it the most difficult of the cases we've discussed. The technique here is to draw the sectional diagram — a sketch of the Earth cut through the poles and through both meridians — and then, by inspection, decide which pole the great circle passes closest to. If you draw the angles reasonably accurately, you can see that the shortest distance between Tokyo and Rio is again via the North Pole, and you base the calculation on that route. So let's break that route into stages. Travelling to the North Pole from Tokyo is an angular distance of 90° minus 35°57', which is 54°03' north. From the North Pole down to the Equator is a further 90°, but now heading south. Then from the Equator down to Rio is 22°10' south. Adding those three stages together — 54°03' plus 90° plus 22°10' — gives a total angular distance of 166°13'. And to convert that angular distance to a linear distance, we use the rule that one minute of arc on a great circle equals one nautical mile. So 166°13' becomes (166 × 60) + 13, which is 9973 nautical miles. Now here's the important check. What if you'd made the initial decision to route via the South Pole instead? Let's see what that would give. From Tokyo down to the South Pole you'd have 35°57' south, then a further 90° south from the Equator to the South Pole, then from the South Pole back up to Rio you'd have 90° minus 22°10', which is 67°50' north. Adding those gives 193°47'. Because that angular distance is greater than 180°, this solution is the longer way around the Earth — it's the major arc, not the minor arc. And here's the elegant part: there's no need to recalculate the whole thing. You simply subtract the longer angular distance from 360° to give the correct answer. So 360° minus 193°47' equals 166°13', which is exactly what we got before. Alternatively, you could continue the wrong calculation to get a linear distance of (193 × 60) + 47, which is 11,627 nautical miles, and then subtract that from the circumference of the Earth, which is 21,600 nautical miles. So 21,600 minus 11,627 gives you 9973 nautical miles — the same answer. And remember, if you do go down that wrong route, you also have to change the directions involved, because the bearing of the great circle will be different on the major arc. Now, Example 5 is a much simpler special case — two points on the Equator. The question is the shortest distance between Dakar, at 00°00' North, 016°35' West, and Singapore, at 00°00' North, 103°55' East. This example uses the fact that a minute of longitude equals one nautical mile, but only at the Equator. That's a crucial qualifier — that conversion only holds at the Equator, because that's the only parallel that is a great circle. The sectional diagram here is drawn in the plane of the Equator, viewed from above the North Pole, and for reference we draw in the Prime Meridian at 00° East/West and the Anti-Prime Meridian at 180° East/West. The change of longitude from Dakar to Singapore is 016°35' in an easterly direction to the Prime Meridian, and then a further 103°55' in an easterly direction to Singapore, giving a total easterly change of longitude of 120°30'. And because we're at the Equator only, that angular measurement of 120°30' equals a linear measurement of (120 × 60) + 30, which is 7230 nautical miles. So the key takeaways from these two examples: first, when longitudes are opposite signs and sum to 180°, you're on a meridian and anti-meridian pair, and you must decide which pole the great circle passes near — and if your first guess gives an angular distance over 180°, you've found the long way around, so subtract from 360°. And second, the minute-of-longitude-equals-one-nautical-mile conversion is valid only at the Equator.