Convergency and Conversion Angle — Page 256, Lesson 229

I want to walk you through the concept of conversion angle, which is one of the most important ideas in general navigation. It's closely tied to convergency, which we'll define as we go. Let's start with a specific situation. Imagine we have two meridians that are 40 degrees of longitude apart. On the Earth, meridians actually converge — they meet at the poles — but for this first diagram, I'm going to show them as parallel lines, like on a Mercator projection. That's just to make the geometry clearer. Now, consider two points: point A at latitude 30° North, longitude 20° West, and point B at latitude 30° North, longitude 20° East. So they're on the same parallel of latitude — 30° North — and they're separated by 40 degrees of longitude. Between A and B, we have two different paths: a Rhumb Line and a great circle. Let's look at the Rhumb Line first. The Rhumb Line between A and B is simply the parallel of latitude at 30° North. That means it has a constant track direction of 090° True — due east — at every point along it. It never changes. The great circle, however, does change direction. The diagram shows that the great circle leaves point A on a track angle of less than 090° True, and it crosses point B on a track angle of more than 090° True. So its direction changes as it goes. We can calculate exactly how much it changes using the formula for convergency. Convergency is the amount by which two meridians converge between two points. The formula is: convergency = change of longitude × sine of the mid-latitude In our example, the change of longitude is 40°, and the mid-latitude is 30° North. So: convergency = 40° × sine 30° Now, sine 30° is 0.5. So: convergency = 40° × 0.5 = 20° That means that whatever track angle the great circle left A on, its track angle at B will be 20° greater. Now, the diagram is completely symmetrical about the mid-longitude — which in this case is 0° East/West. We know the Rhumb Line track is 090° True all the way, and that is also the direction of the great circle track at the mid-longitude point. So, because the great circle changes by 20° total, and it's symmetrical, it must have left A on a track of 080° True and arrived at B on a track of 100° True. That's 10° less than the Rhumb Line at A, and 10° greater at B. That brings us to the definition of conversion angle. Conversion angle is the difference between the great circle direction and the Rhumb Line direction joining two given points. In our example, conversion angle is 10°. At point A, it's the difference between 080° True (the great circle track) and 090° True (the Rhumb Line track). At point B, it's the difference between 100° True (the great circle track) and 090° True (the Rhumb Line track). And importantly, conversion angle is always the same at each end of the two points connected by the great circle and the Rhumb Line. This leads to two key relationships. First, conversion angle is half of convergency. So if convergency is 20°, conversion angle is 10°. Second, the formula for conversion angle is: Conversion angle = ½ × change of longitude × sine of the mean latitude In our example, that's ½ × 40° × sine 30°, which is 20° × 0.5 = 10°. Exactly what we found. Now, I want to mention one more thing. It's equally possible to draw this situation with the meridians shown as convergent — as they really are on the Earth. In that case, the great circle appears as a straight line, even though its track angle changes direction, and the Rhumb Line appears as a curve, even though it's a line of constant True direction. But the key fact remains: the great circle always runs nearer to the nearer pole, exactly as it does on the globe. So to summarise: convergency is the total angular change between two meridians between two points. Conversion angle is half of that, and it's the difference between the great circle track and the Rhumb Line track at either end. That's the core idea you need to hold onto.