Convergency and Conversion Angle — Page 247, Lesson 221

Let's start with the definition itself, because everything in this chapter hangs on it. Convergency — also called Earth Convergence — is defined as the angle of inclination between two selected meridians, measured at a given latitude. Let me unpack that. Imagine two meridians — lines of longitude — running from the North Pole down to the South Pole. On a flat map they look parallel, but on the actual spherical Earth they are not parallel at all. They start together at the Pole, spread apart at the Equator, and come back together at the other Pole. The "angle of inclination" is simply how much those two meridians lean toward each other at a particular latitude. That leaning is convergency. Now, the key relationship: the amount of convergence depends on where you are. Look at the two extremes. At the Equator, latitude is 0°, and the ratio of convergency to change of longitude is zero — the meridians are parallel there, so no convergence at all. At the Pole, latitude is 90°, and that ratio is a factor of one — the meridians meet at the Pole, so the convergence is exactly equal to the change in longitude. So the relationship between convergency and latitude is a function of the sine of the latitude. That gives us the formula: Convergency = Change in Longitude × Sine Latitude. Let me make that concrete with the example in the figure. Suppose we pick latitude 30°N. We draw tangents to each meridian at that latitude — straight lines touching the curved meridian at that point. The angle between the slopes of those two tangents is the convergency. In the example, the change of longitude is 40°, and sine of 30° is 0.5. So convergency equals 40° times 0.5, which is 20°. So now we have a working formula for convergency between any two meridians at the same latitude. But you might be wondering — why does a pilot care? That's the crucial question, and the chapter answers it directly. Here's the reason. At any point on Earth, True North is defined with reference to the direction of the local meridian. If you're flying and you move to a different meridian than the one you started on, your local direction of True North has changed. That's exactly why a rhumb line is a continuously curved line on the Earth's surface. A rhumb line cuts all meridians at the same angle — but since the meridians themselves are at different angles to each other, the line has to keep curving to maintain that constant angle. Now contrast that with a great circle track. All automatic computing systems — Inertial Navigation Systems, Global Positioning Systems, and Flight Management Computers — compute steering signals along great circle tracks. When you fly a great circle, your track direction will alter over time with respect to True North by exactly the amount of convergency. Why? Because the track direction is defined by reference to the meridians. If the direction of the meridians changes, then the track direction changes. So the change in great circle track direction is precisely the angle of inclination of the meridians at the two points where the track is measured. That's the practical link — convergency isn't just a geometric curiosity; it's the amount by which your great circle heading changes as you move across meridians. Let me show you the geometry with the figures. shows maximum convergence at the Pole. shows intermediate convergence at a mid-latitude. shows converging and diverging meridians. shows the convergency angle with the tangents drawn. And shows a great circle cutting two meridians, X and Y, with tangents to the meridians drawn — that's the picture of why the track direction changes. So to tie it all together: convergency is the angle between two meridians at a given latitude, computed as change in longitude times sine of latitude. At the Equator it's zero; at the Pole it equals the change in longitude. And for a pilot, this matters because your great circle track direction changes relative to True North by exactly the convergency — because True North is defined by the local meridian, and the meridians converge.