Let's pick up right where the mathematics left off — Page 367, Lesson 324

Let's pick up right where the mathematics left off. We just proved that between latitudes 90° and 78°, the scale on the polar stereographic chart stays within 1% of the scale at the Pole. That's a big deal, because it means we can treat that whole zone as a constant-scale chart. The numbers we saw — 0.989074, 989,074, 1,000,000 — those are the ratios showing how close the scale is to perfect. So out to 78° North, we're within 1%. If we push further out to latitude 70°, the scale is still within 3% of the Pole's scale. Let me summarize that cleanly: between 90° and 78°, scale is within 1% of the Pole; between 78° and 70°, scale is between 1% and 3%. Now, why does this constancy matter so much? Because it makes the polar stereographic projection almost ideal for flying in polar regions. Think about the history here. With the breakdown of former political rivalries, and the arrival of long-range, high-endurance aircraft, ETOPS procedures, and inertial navigation systems, polar routes have become common. These routes are often shorter, and some go very close to the Pole indeed. Let me put some real numbers on this. Take the 1% scale zone. The co-latitude of 78° is 12°. Co-latitude is just 90° minus the latitude, so 90 minus 78 gives us 12°. The distance from the Pole to latitude 78° is 60 nautical miles times 12, which is 720 nautical miles. Now, if an aircraft enters the 1% zone on one side and leaves more or less on the opposite side of the circle, the total distance inside that zone is around 1,400 nautical miles. At typical jet-liner speeds of around 450 knots true airspeed, that's approximately 3 hours of flying time. So a chart that can be considered constant scale, allowing you to measure distances with a ruler for over 3 hours of flying, is a very useful projection indeed. Now let's talk about orthomorphism. On this projection, meridians are straight lines originating from the pole. Parallels of latitude are arcs of circles centred at the pole. The expansion happens at the same rate in any direction from any point. That's the definition of orthomorphic — the projection preserves angles locally. Because the expansion is uniform in all directions, the projection is orthomorphic. The graticule — that's the network of meridians and parallels — follows the same pattern: meridians are straight lines from the pole, and parallels are arcs of circles centred at the pole. But here's the catch with shapes. Because of scale expansion, shapes and areas will be distorted away from the pole. So while angles are preserved locally, the overall shapes and sizes of features get stretched as you move away from the Pole. Now, chart convergence. This is a subtle but crucial point. The meridians converge at the pole at exactly the same rate as they do on the Earth's surface. So on the projection, the convergency remains the same, whereas on Earth it decreases away from the Pole. Let me unpack that. On the actual Earth, meridians converge at the Pole, but as you move away from the Pole, the rate of convergence decreases. On this chart, however, the convergence stays constant everywhere. This gives the polar stereographic chart a convergency factor, or 'n', of 1. The chart convergence is always exactly equal to the change of longitude between two points, whatever the latitude. So if you have two points with a 30° change of longitude, the chart convergence is 30°, no matter where those points are. Chart convergence is constant all over the chart. But — and this is the key limitation — chart convergence is 'correct', meaning the same as on Earth, only at the Pole itself. So to tie it all together: we have a chart that's constant scale within 1% out to 78° North, orthomorphic because expansion is uniform in all directions, with a convergency factor of 1, meaning chart convergence always equals change of longitude. That's why it's the go-to projection for polar navigation.