Convergency and Conversion Angle — Page 256, Lesson 229

Let’s start with the big picture. On the Earth, meridians are not parallel—they converge, meaning they get closer together as you move toward the poles. That convergence is the whole reason a great circle and a rhumb line behave differently. I want to walk you through the idea of conversion angle, which is the difference between those two directions. First, let’s set up the example. We have two meridians that are 40 degrees of longitude apart. We’re looking at a route from point A at 30°N, 020°W, to point B at 30°N, 020°E. So both points are on the same parallel of latitude, 30° north, and they’re separated by 40 degrees of longitude. Now, consider two possible tracks between A and B. The rhumb line is a line of constant true direction. In this case, because both points lie on the parallel of 30°N, the rhumb line is simply that parallel, and its track is constant at 090°(T)—due east—at every point along it. That’s the definition of a rhumb line: it crosses every meridian at the same angle. The great circle, on the other hand, changes direction continuously. The diagram shows it leaves A on a track angle of less than 090°(T), and it crosses B on a track angle of more than 090°(T). So the great circle is curving northward relative to the rhumb line, because on the globe the great circle runs nearer the pole. Now, how much does the great circle change direction between A and B? That’s where convergency comes in. Convergency is the amount by which the meridians converge between two points, and it’s given by this formula: convergency equals change of longitude times sine of the mid-latitude. In symbols, convergency = ch.long × sine mid lat. Let’s plug in our numbers. Change of longitude is 40 degrees. Mid-latitude is 30 degrees, and sine of 30 degrees is 0.5. So convergency = 40 × 0.5 = 20 degrees. That means the great circle’s track angle at B is 20 degrees greater than whatever it was at A. Now, here’s the clever part. The diagram is completely symmetrical about the mid-longitude, which in this example is 0°E/W. At that mid-longitude, the great circle’s track is exactly the same as the rhumb line’s track—090°(T). So if the great circle gains 20 degrees of track angle from A to B, and it’s 090°(T) at the middle, then it must have left A on a track of 080°(T) and passed through B on a track of 100°(T). Half of the 20-degree convergency on each side. That brings us to the key definition. Conversion angle is the difference between great circle direction and rhumb line direction joining two given points. In our example, at A, it’s the difference between 080°(T) and 090°(T), which is 10 degrees. At B, it’s the difference between 100°(T) and 090°(T), also 10 degrees. Notice that conversion angle is always the same at each end of the two points connected by the great circle and the rhumb line. So we have two formulae to remember. First, conversion angle equals half of convergency. Second, conversion angle equals half of change of longitude times sine of the mean latitude. In symbols, conversion angle = ½ ch.long × sin mean lat. In our example, that’s ½ × 40 × sine 30° = 20 × 0.5 = 10 degrees. Now, one more thing about how we draw this. The first diagram was drawn as a Mercator-type projection, with meridians shown as parallel lines, even though they actually converge on the Earth. That was done to make it clear that the rhumb line track does not change track angle, but the great circle does. However, it’s equally possible to draw the same situation with convergent meridians. 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 regardless of the projection, the great circle still runs nearer the pole, exactly as it does on the globe. So the takeaway is this: convergency is the total angular change of the meridians between two points, and conversion angle is half of that—the difference between the great circle and rhumb line directions at either end. That’s the core of this section.