Loxodromes, rhumb lines, and the problem of sailing into infinity at the North Pole

Want to know how to get from New York to London? Set a course 73 degrees east of north and just keep going. Congratulations, you're on the express loxodrome.

Loxodromes are also known as rhumb lines. A rhumb is a path that will take you from one point of the earth to another if you maintain a constant bearing. They are not the fastest ways to get anywhere, but they were a boon to early sailors who didn't have access to navigational equipment. Rhumb lines allowed sailors to keep a ship's bow pointed at the right angle and go.

Rhumb lines hit each longitudinal line on earth at the same angle — hit the Prime Meridian at Greenwich at 40 degrees east of north and you'll hit the next line at the same angle. But the longitudinal lines pinch together at the poles, forcing the rhumb line into a tighter and tighter curve. The largest curves — those that span the globe and go into endless spirals as you get closer to the poles — are the loxodromes.

Loxodromes, rhumb lines, and the problem of sailing into infinity at the North Pole

Or at least they appear as curves on a globe. Early sailors didn't particularly want to plot an endless spiral, so these curves had to be flattened out and made into simple lines. To achieve this untangling, mapmakers devised a new way to depict the earth: the Mercator map.

The Mercator layout mangled distances and butchered continents. And though the distances between points were changed, the angles between points on a Mercator are spot-on. Overland distances meant nothing to sailors; of greater significance was the ease of charting a course. On the Mercator, all a sailor had to do was set down a line between his current location and destination and stick with the required bearing. Nowadays, the Mercator is infamous for giving generations of school children a massively distorted sense of the world. And it's all thanks to the inherent difficulty of the spiral.

Via Wolfram Mathworld and Case Western (PDF). Top Image Geographicus. Second Image: Maksim/Wiki Commons.