Rhumb Line Distance

A rhumb line — a loxodrome — is the track that crosses every meridian at the same angle. It is what you get by setting one course and holding it, which is why it was the working navigator's line for four hundred years and why it is still the line drawn between two waypoints on a plotter. On a Mercator chart it is a straight line, and that is not a coincidence: Mercator built his projection in 1569 for exactly this purpose, stretching the latitude scale precisely enough that a constant bearing plots straight.

That stretching is the $q$ in the formula. The Mercator or "stretched" latitude $\psi = \ln\tan(45^{\circ} + \varphi/2)$ is how far up the chart a given latitude falls, and $q = \Delta\varphi/\Delta\psi$ is the ratio of real latitude change to charted latitude change over the leg. When the two latitudes are close, $q$ tends to $\cos\varphi$, and the whole expression collapses into the old plane-sailing rule that departure equals difference of longitude times the cosine of the middle latitude. The general form simply handles the case where the leg is long enough that the cosine has moved.

The price of a single course to steer is distance, and on this shard's anchor pair — New York to London — it is about 118 nautical miles out of 3 109, just under four percent. That gap is what great circle sailing exists to recover. It grows with latitude and with the east-west span of the leg: it is negligible near the equator, negligible on any north-south leg, and enormous on a North Atlantic or North Pacific crossing. Sailing a rhumb line across an ocean because it is easier, without knowing what it costs, is the mistake this number exists to prevent — and so is the opposite mistake of computing great circles for a twenty-mile hop where the two answers are identical to within the width of the pencil.

Two edge cases are worth knowing. A rhumb line to a pole has infinite length: the Mercator latitude runs to infinity at 90°, and a loxodrome heading for the pole spirals in around it forever, getting closer and never arriving. And a rhumb line running due east or west along a parallel is a perfectly good track that is not a great circle at all — which is why a plane flying "straight west" along the 60th parallel is taking a noticeably longer route than the one that bulges north.