Great Circles, Rhumb Lines & Directions on the Earth — Page 28, Lesson 33

Let’s start with the most important idea in this whole chapter: the Great Circle. I want you to picture the Earth as a perfect sphere for now. A Great Circle is a circle drawn on the surface of the Earth whose centre and radius are exactly those of the Earth itself. In other words, the circle’s centre is the Earth’s centre, and its radius is the Earth’s radius. That’s the precise definition. Why is it called “great”? Because if you cut a disc through the Earth in the plane of that Great Circle, that disc would have the largest area you can possibly achieve. Any other circle you draw on the surface — one that doesn’t pass through the Earth’s centre — would give you a smaller disc. So “great” means the biggest possible cross-section. Now, here’s the key operational fact: the shortest distance between any two points on the Earth’s surface is the shorter arc of the Great Circle joining those two points. So if you want to fly the absolute minimum distance between two airports, you follow the Great Circle track. And there’s a subtlety: given two points on the Earth’s surface, there is only one Great Circle joining them — unless the two points are diametrically opposed, meaning they’re exactly opposite each other on the globe, like the North and South Poles. In that special case, you have infinitely many Great Circles passing through both. Let me show you what that looks like. Now, contrast that with the Rhumb Line. A Rhumb Line is a regularly curved line on the surface of the Earth which cuts all meridians at the same angle — it’s a line of constant direction. Think of it this way: as you travel along a Rhumb Line, your track angle relative to true north stays the same at every meridian you cross. That’s why it’s called a line of constant direction. This property was enormously important for mariners and aviators until about 40 years ago, when cheap, powerful computing became widely available. Before that, you would calculate a constant straight-line track and then hold the compass heading that would give you that track. That made establishing the track a relatively simple problem. But here’s the trade-off: unlike the Great Circle route, the Rhumb Line does not give the shortest distance over the Earth between the two points. Let me show you a real example. This is the Moscow to Vancouver route. You can see the Rhumb Line track curving across the map. Now compare that with the Great Circle track between the same two points. The Great Circle passes very close to the North Pole, and it is much shorter. That’s the whole point — the Great Circle gives you the shortest distance, but it’s harder to fly because your heading changes continuously. The Rhumb Line is easier to fly because you hold one constant heading, but you pay for that simplicity with extra distance. So to summarise what we have: a Great Circle is the shortest path and the largest possible circle on the Earth’s surface, but it requires constant heading changes. A Rhumb Line cuts all meridians at the same angle, giving you a constant direction, but it’s not the shortest distance. In the Moscow–Vancouver case, the Great Circle hugs the North Pole and is much shorter than the Rhumb Line. That’s the fundamental trade-off you’ll be working with throughout navigation.