Convergency and Conversion Angle — Page 247, Lesson 223

Let’s pick this up right where the geometry gets interesting. We’ve already seen how meridians converge at the poles and how we measure that convergence. Now I want to walk you through the precise definition of convergency, because this is the heart of the whole chapter. Look at Figure 14.7. We have a great circle cutting two meridians, which I’ll call meridian X and meridian Y. At the point where the great circle crosses meridian X, I draw a tangent to that meridian. At the point where it crosses meridian Y, I draw a tangent to that meridian. The angle between those two tangents is the convergency — exactly the same idea we used before, just applied at two different meridians. Now, here’s the key. The great circle direction at meridian X is represented by the angle ‘a’. The great circle direction at meridian Y is represented by the angle ‘b’. The difference between them — that is, ‘b’ minus ‘a’ — equals the convergency. So convergency is literally the change in great circle direction as you travel from one meridian to the next. But here’s the catch, and this is the subtle part. In this case, the great circle does not cross the two meridians at the same latitude. In fact, there are only two mean track directions where it will cross at the same latitude: due East or due West. On any other mean track, by definition, there will be some element of north or south latitude change between the two meridians. So why does that matter? Because our simple convergency formula — the one we used before — requires a single term for ‘sine latitude’. When the great circle crosses at different latitudes, there is no single latitude to plug in. Instead, the calculation requires the sine of the mean latitude between the two points. Now, determining the mean latitude is a complex process, because the mean latitude is closer to the nearer pole than the mid-latitude is — with two exceptions. So although the formula strictly requires mean latitude, for simplicity we use mid-latitude instead. And here’s the good news: the difference between the sine of mid-latitude and the sine of mean latitude is so small that it is virtually insignificant — unless a large change of longitude is involved, which is unlikely in practice. So let me give you the full, precise definition, because this is what you need to remember. Convergency is the angle of inclination between two selected meridians, measured at a given latitude. And it is equal to the difference between the great circle directions measured at each meridian. And the formula we use to calculate it is this: Convergency = Change in Longitude × Sine Mean Latitude. Let me unpack that. Change in Longitude is the difference in longitude between the two meridians you’re considering. Sine Mean Latitude — and remember, we’re using mid-latitude as a practical approximation for mean latitude — is the sine of the latitude at which you’re measuring. Multiply those two together, and you get the convergency in degrees. So the whole idea is: the closer you are to the pole, the larger the sine of the latitude, and therefore the greater the convergency for a given change in longitude. At the equator, the sine is zero, so there’s no convergency at all. That’s the full picture — the definition, the formula, and why we use mid-latitude instead of mean latitude.