What is the initial great circle track from B back to A — Page 247, Lesson 225

I want to walk you through a worked example that shows how to handle a typical exam question on great circle tracks and convergency. This is a practical application of what we've been building up to. The question is this: The initial great circle track from A, at 40° North, 002° West, to B, at 50° North, 010° East, is 060° True. What is the initial great circle track from B back to A? The first thing I want you to do is draw a sketch diagram. It doesn't need to be a scale drawing — just a rough sketch to get the geometry clear in your mind. Since we're in the Northern hemisphere, draw a couple of meridians that converge as they go northwards. That's the key shape: lines of longitude getting closer together as you go toward the North Pole. Now, we know the initial great circle track from one of these points is 060° True. I want you to make that the left-hand meridian. Why? Because if you draw the 060° track from the right-hand meridian, the bearing arrow would go off the edge of your diagram. So choose the left-hand one. That means the left-hand intersection of the track with a meridian must be point A, and the right-hand one must be point B. The question tells us the initial great circle track from A is 060° True, so that fits. Let's check this makes sense using the actual latitudes and longitudes. A is at 40° North, 002° West. B is at 50° North, 010° East. With an initial track of 060° True from A, B should be north of A — and it is, because 50° North is north of 40° North. B should also be east of A — and it is, because 010° East is east of 002° West. So the diagram is correct. Now we can calculate the convergency. The formula is: Convergency equals Change in Longitude times Sine Mean Latitude. First, the change in longitude. We go from 002° West to 010° East. That's a total change of 12 degrees. If you're not sure: from 002° West to 0° is 2 degrees, then from 0° to 010° East is another 10 degrees, so 2 plus 10 equals 12 degrees. Next, the mean latitude. That's the average of the two latitudes: 40° North and 50° North. Halfway between them is 45° North. The sine of 45 degrees is 0.7071. So convergency equals 12 degrees times 0.7071. That gives us 8.5 degrees. Now, what does this convergency value tell us? It's the angle between the tangents to the meridians at A and B. And it's the key to finding the reciprocal track. The initial great circle track from B to A will be the initial track from A to B, plus or minus the convergency, depending on direction. In the Northern hemisphere, when going from west to east, the great circle track increases. So from A to B we had 060° True. The track from B back to A would be 060° plus 180° for the reciprocal, then adjusted by the convergency. But let's be precise: the initial great circle track from B to A equals the initial great circle track from A to B, plus 180°, minus the convergency. So that's 060° plus 180° equals 240°, minus 8.5° gives 231.5° True. That's how you work through this type of question step by step: sketch the converging meridians, place your points correctly, calculate the convergency using change in longitude times sine mean latitude, then apply it to find the reciprocal great circle track.