The Polar Stereographic Chart — Page 367, Lesson 326

Let’s pick this up right where the geometry gets interesting. We’ve got the Polar Stereographic chart, and now we’re looking at how three different lines behave on it: the Rhumb Line, the Great Circle, and a plain straight line drawn on the chart. I want you to picture Figure 23.4. On that chart, we’ve drawn all three lines between two longitudes — 000°E/W, which is the Greenwich Meridian, and 090°E — and we’ve done this at two different latitudes: 50°N and 70°N. So we’re comparing how these lines curve at a lower latitude and a higher latitude. Now, the first thing to notice is the Rhumb Line. As with the Lambert chart, the Rhumb Line has the greatest curvature of the three. Why? Because a parallel of latitude must cut meridians at right angles. That’s the defining property of a Rhumb Line — it crosses every meridian at the same angle. And on this chart, the Rhumb Line is always concave to the pole of projection. Concave means it curves inward, bowing toward the pole. So the Rhumb Line bends toward the pole of projection. Next, the Great Circle. Again, as with the Lambert chart, the Great Circle is concave to the parallel of origin. But here’s the key difference: on the Polar Stereographic chart, the parallel of origin is the pole itself. So the Great Circle is also concave to the pole of projection — it bends toward the pole too — but its curvature is less than the Rhumb Line’s, in the same hemisphere. So both curve toward the pole, but the Rhumb Line curves more. Now, here’s the practical question: how closely does a straight line drawn on the chart approximate a Great Circle? Because on a chart, a straight line is what you can actually draw with a ruler. We can calculate the difference between a Great Circle and a straight line at any latitude. Here’s the relationship to remember. The angle between the straight line and the Rhumb Line is half chart convergence. And the angle between the Great Circle and the Rhumb Line is the conversion angle, which is half Earth convergence. So we have two different convergence angles: chart convergence and Earth convergence. Let me make that concrete with the numbers from the excerpt. At 50°N, over a 90° change of longitude, the angle between a straight line and a Rhumb Line is half chart convergence. For a Polar Stereographic chart, the chart convergence factor ‘n’ equals 1. So over 90° of longitude, chart convergence is 90°, and half of that is 45°. That’s the angle between the straight line and the Rhumb Line at 50°N. Now the angle between the Great Circle and the Rhumb Line is the conversion angle. The formula is: conversion angle equals half of the change of longitude times the sine of the mean latitude. So we take ½ × 90° × sin(50°). The sine of 50° is 0.766. So ½ × 90 × 0.766 gives us 34½°. That’s the conversion angle at 50°N. So at 50°N, the straight line sits 45° away from the Rhumb Line, but the Great Circle sits only 34½° away from the Rhumb Line. That means the straight line and the Great Circle are not the same thing — there’s a real angular difference between them. The straight line is a good approximation, but it’s not exact, and the difference is calculable at any latitude using these two angles. Let me make sure the distinction is crisp. Chart convergence is the angle between meridians as drawn on the chart — on the Polar Stereographic, with ‘n’ = 1, it equals the change of longitude. Earth convergence is the angle between meridians on the actual Earth’s surface, which depends on latitude — that’s where the sine of the mean latitude comes in. Half of each gives you the angle to the Rhumb Line, and the difference between those two angles is how far the straight line departs from the true Great Circle. So the takeaway: on the Polar Stereographic chart, both the Rhumb Line and the Great Circle are concave to the pole of projection, the Rhumb Line curving more. A straight line on the chart is a close but not exact approximation of a Great Circle, and you can quantify the error at any latitude using half chart convergence and conversion angle.