Great Circles, Rhumb Lines & Directions on the Earth — Page 28, Lesson 36

Let’s pick this up right where the idea of a Rhumb Line gets its full definition. We already know a Rhumb Line is that regularly curved line that cuts every meridian at the same angle. Now I want to lock in the key fact: just like with Great Circles, there is only one Rhumb Line that can be drawn between any two points. So for a given pair of places, you get exactly one Great Circle and exactly one Rhumb Line connecting them. Now, the common examples of Rhumb Lines. First, the Parallels of Latitude. They cut all meridians at 90°, so they qualify as Rhumb Lines. Second, the Equator — and this is a special case, because the Equator is also a Great Circle. Third, the Meridians. They are also Great Circles, and the cut angle involved is 0°, because a meridian runs north–south and crosses other meridians at the poles. Here’s a really important relationship to remember: the Rhumb Line between two points will always lie nearer to the Equator than the corresponding Great Circle. Conversely, the Great Circle between those same two points will always lie nearer to the Pole than the Rhumb Line. So if you picture a route between two cities, the Great Circle bows toward the pole, and the Rhumb Line stays closer to the equator. Some explanations describe the Rhumb Line’s shape as convex to the Equator or concave to the nearer Pole — that’s just another way of saying the same thing. Now, which lines are both Great Circles and Rhumb Lines? Only two types: the Equator and any meridian — along with its associated anti-meridian, which is the meridian exactly 180° away. There are no other lines that are both. The parallels of latitude are Rhumb Lines because they cut all meridians at 90°, but they are Small Circles, because they do not have the same radius and centre as the Earth. Now let’s talk about Great Circle direction. The direction of the Great Circle over the Earth’s surface changes as you travel along it — we’ve seen that. And the Great Circle track is always nearer to the nearer pole. Let me walk you through Figure 2.6, which is a Mercator chart showing two Rhumb Line tracks, each in an east–west direction — one in the Northern hemisphere, one in the Southern. Both are at fairly high latitudes, say 50°N or 50°S, to show a reasonable curvature on their Great Circle tracks. But the effect occurs, to some extent, at any latitude other than the Equator. Let’s take the Northern hemisphere first. Consider the track left to right — a Rhumb Line track of 090°, meaning due east. If you’re heading in an easterly direction, the Great Circle track starts with an initial direction of about 030°, then curves round to 090°, and finishes up on about 150°. In other words, the track direction is increasing. Now if you’re heading in a westerly direction along that same Rhumb Line of 090°, the Great Circle track starts with an initial direction of about 330°, then curves round to 270°, and finishes up on about 210°. In other words, the track direction is decreasing. Now the Southern hemisphere. Again, consider the track left to right — a Rhumb Line track of 090°. If you’re heading in an easterly direction, the Great Circle track starts with an initial direction of about 150°, then curves round to 090°, and finishes up on about 030°. In other words, the track direction is decreasing. And if you’re heading in a westerly direction, the Great Circle track starts with an initial direction of about 210°, then curves round to 270°, and finishes up on about 330°. In other words, the track direction is increasing. So the pattern is: in the Northern hemisphere, an easterly Great Circle track increases in direction, and a westerly one decreases. In the Southern hemisphere, it’s the opposite — an easterly Great Circle track decreases, and a westerly one increases. That’s the core of how Great Circle direction behaves relative to the Rhumb Line.