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 in front of us, and now we're looking at how lines behave on it — specifically the difference between a Rhumb Line, a Great Circle, and a straight line drawn on the chart. Look at Figure 23.4. 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 versus a higher one. Now, just like we saw with the Lambert chart, the Rhumb Line has the greatest curvature. Remember, a Rhumb Line is a line that crosses every meridian at the same angle. And here's the key property: a parallel of latitude must cut meridians at right angles. That's a geometric necessity — parallels and meridians are always perpendicular to each other. So the Rhumb Line, which follows a constant bearing, curves in a very specific way. And the rule on this chart is: Rhumb Lines are always concave to the pole of projection. Concave means the curve bows inward, toward the pole. So if you're looking at a Rhumb Line, its curve always bends toward the pole of projection. Now, the Great Circle. Again, like the Lambert chart, the Great Circle is concave to the parallel of origin. But here's the twist on the Polar Stereographic chart: the parallel of origin is the pole itself. The pole is the point of tangency — that's where the flat surface touches the globe. So on this chart, Great Circles are also concave to the pole of projection. But — and this is important — the amount of curvature is less than the Rhumb Line in the same hemisphere. So the Rhumb Line curves more, the Great Circle curves less, and a straight line drawn on the chart curves not at all. Now, here's where we can actually quantify this. It's possible to calculate the difference between a Great Circle and a straight line at any latitude, to see how closely a straight line approximates a Great Circle. And the key relationships are these: the angle between the straight line and the Rhumb Line is half chart convergence. Meanwhile, the angle between the Great Circle and the Rhumb Line is the conversion angle, which is half Earth convergence. Let me make sure that distinction is crystal clear, because it's the heart of this page. Chart convergence is the angle between the meridians as drawn on the chart — it's a property of the chart projection. Earth convergence is the angle between the meridians on the actual globe — it's a property of the Earth itself. Half chart convergence tells you how much the straight line deviates from the Rhumb Line. Half Earth convergence — that's the conversion angle — tells you how much the Great Circle deviates from the Rhumb Line. Now let's plug in real numbers 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 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 conversion angle — the angle between the Great Circle and the Rhumb Line. The formula is: conversion angle equals ½ × ch.long × sin mean latitude. So we take half of the change of longitude — half of 90° is 45° — and multiply by the sine of the mean latitude. At 50°N, the sine of 50° is 0.766. So we get ½ × 90 × 0.766, which works out to 34½°. That's the conversion angle at 50°N. So now you can see the full picture. At 50°N, over 90° of longitude: the straight line sits 45° away from the Rhumb Line, but the Great Circle sits only 34½° away from the Rhumb Line. The straight line is a closer approximation to the Great Circle than it is to the Rhumb Line — but it's still not exactly the Great Circle. The difference between the straight line and the Great Circle is the gap between those two angles: 45° minus 34½°, which is 10½°. That's the error you'd introduce if you flew a straight line on the chart thinking it was a Great Circle. And remember, this is at 50°N. The same exercise at 70°N would give you different numbers, because the sine of the mean latitude changes — the conversion angle changes with latitude. So the lesson here is: on the Polar Stereographic chart, a straight line is a good approximation to a Great Circle, but it's not exact, and the error is calculable at any latitude using these relationships.