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

I want to walk you through the relationship between Great Circles and Rhumb Lines, and then how Great Circle direction behaves across the Earth’s surface. Let’s start with an important point: just like with Great Circles, there is only one Rhumb Line that can be drawn between any two points. That’s a key property — unique, not multiple. Now, what are some common examples of Rhumb Lines? The first is parallels of latitude — those east-west circles around the Earth. They are Rhumb Lines because they cut all meridians at a constant angle of 90°. Next, the Equator is a special case: it is a Rhumb Line, but it is also a Great Circle. Finally, meridians are also Rhumb Lines — and they are Great Circles too. The cut angle involved for a meridian is 0°, meaning they run north-south and cross all other meridians at that constant angle. Here’s a useful rule 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. Some explanations describe the shape of the Rhumb Line 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: the Equator and any meridian (along with its associated anti-meridian — the meridian exactly 180° opposite). No other lines qualify. The parallels of latitude, as I mentioned, are Rhumb Lines because they cut all meridians at 90°, but they are Small Circles — they do not have the same radius and centre as the Earth. Let’s move to Great Circle direction. The direction of a Great Circle over the Earth’s surface changes continuously, as we’ve seen. The Great Circle track is always nearer to the nearer pole. Let’s look at Figure 2.6, which shows a Mercator chart with two Rhumb Line tracks, each running east-west — one in the Northern hemisphere and 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 a track left to right — a Rhumb Line track of 090° (due east). If you are 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 as you go. If you are heading in a Westerly direction along that same Rhumb Line of 090° (but going right to left), the Great Circle track starts with an initial direction of about 330°, then curves round to 270°, and finishes up on about 210°. Here, the track direction is decreasing. Now the Southern hemisphere. Again, consider a track left to right — a Rhumb Line track of 090°. If 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 this case, the track direction is decreasing. If 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°. Here, the track direction is increasing. So the pattern flips between hemispheres: in the Northern hemisphere, an easterly Great Circle track increases in direction; in the Southern hemisphere, an easterly Great Circle track decreases. And the opposite holds for westerly tracks. That’s the behaviour you need to understand for navigation planning.