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

I want to walk you through two more worked examples that build directly on what we just covered — calculating the shortest distance between two points on the same meridian, and then on a meridian and its anti-meridian. These are classic exam-style problems, so let's take them step by step. Example 2 — Same meridian, different hemispheres We have Durban, position D, at 29°30' south, 030°30' east, and Leningrad, position E, at 59°47' north, also 030°30' east. Both are on the same meridian — same longitude — but they're in opposite hemispheres: one south, one north. Because they're on the same meridian, the shortest path between them is simply along that meridian, over the Equator. The angular distance is the change of latitude, or ch.lat, which here is the sum of the two latitudes since they're on opposite sides of the Equator. So we add 29°30' south and 59°47' north. That gives us 89°17' of angular distance north from D. Now, to convert that angular distance into a linear distance in nautical miles, we use the standard rule: one minute of arc along a meridian equals one nautical mile. So 89 degrees is 89 × 60 = 5340 minutes, plus the remaining 17 minutes gives us 5357 nautical miles. That's the shortest distance between Durban and Leningrad. Example 3 — Meridian and anti-meridian, same hemisphere Now 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 northern hemisphere, but they are on a meridian and its anti-meridian. What does that mean? The two longitude values are of opposite sign — one east, one west — and when you add them numerically, they total 180 degrees. Let's check: 011°10' east plus 168°50' west equals exactly 180 degrees. So Rome and Honolulu lie on opposite sides of the globe, on the same great circle that goes through the poles. This is a surprising but important result: the great circle between Rome and Honolulu goes over the North Pole. If you wanted to fly that route in one stage, you'd head north, over the Pole, then south to Honolulu. To calculate the angular distance, the simplest method is to add the two latitudes and subtract their total from 180 degrees. So 41°55' plus 21°17' equals 63°12'. Subtract that from 180 degrees gives us 116°48' of angular distance. An alternative method is to calculate the change of latitude from Rome to the North Pole: 90 degrees minus 41°55' equals 48°05'. Then from the North Pole to Honolulu: 90 degrees minus 21°17' equals 68°43'. Add those two together: 48°05' plus 68°43' also gives 116°48'. Both methods are mathematically the same — just two ways of arriving at the same answer. Now convert that angular distance to nautical miles: 116 degrees times 60 equals 6960 minutes, plus the remaining 48 minutes gives us 7008 nautical miles. Finally, the initial direction from Rome to the Pole is north, and then from the Pole to Honolulu it's south. So the great circle track starts north, crosses the Pole, and then heads south.