The Polar Stereographic Chart — Page 367, Lesson 322

I want to walk you through the Polar Stereographic Chart now. This is a big one, because it's the chart you'll actually use for polar navigation, and it behaves differently from the other projections you've seen. First, the headline: of all the projections we've discussed, the polar stereographic is the only geometric projection — meaning it's a true perspective projection. That's a precise term. A geometric or true perspective projection means the chart is literally drawn by projecting points from the Earth's surface onto a flat plane along straight lines, as if from a light source. It's not a mathematical compromise; it's a physical construction. Let me set up that construction, because it explains everything that follows. Look at Figure 23.1. We take a flat surface and place it so it just touches the Earth at the North Pole. That point of contact is called the point of tangency. Then we position the light source at the South Pole — which is diametrically opposed to the North Pole, meaning exactly opposite on the globe. The light shines from the South Pole, through the Earth's surface, and casts shadows of the graticule — that's the network of meridians and parallels — onto the flat surface. That's the geometrical projection, and you can see the resulting graticule in the lower part of the diagram. Now, what does that graticule look like? This is the key visual. The meridians — the lines of longitude — are straight lines radiating outward from the Pole. The parallels — the lines of latitude — are concentric circles, all centred on the Pole. But here's the crucial detail: those parallels are not evenly spaced. Their spacing from the Pole depends on something called the scale expansion from the Pole. The further you get from the Pole, the more the scale stretches, and that's why the circles spread further apart. To work out exactly how they spread, we need a new concept: the co-latitude. This is simply 90° minus the latitude. So if the latitude is 60°, the co-latitude is 30°. Keep that definition in your head — it's the foundation for the whole scale formula. Now let's trace the light paths in Figure 23.1 to see why half the co-latitude keeps appearing. Take the light ray that passes through the Equator. The Equator's latitude is 0°, so its co-latitude is 90°. Half of that co-latitude is 45°. Now look at the triangle formed by that light path. The opposite side and the adjacent side are both the radius of the Earth — equal lengths — so we have a right-angled triangle with a bottom angle of 45°. That's the geometric reason the half co-latitude shows up. There's a geometrical proof that the angle the light path makes with the axis of the Earth is always a function of half the co-latitude. You are not required to understand that proof — I want you to remember the result, because that's what you'll use. The result is this: the scale expands away from the Pole at a rate equal to the secant squared of half the co-latitude. So the formula is: scale expansion = sec² (½ co-latitude). Let me make sure that's clear. Secant is the reciprocal of cosine — sec of an angle is 1 divided by the cosine of that angle. And we're squaring that whole thing. So at the Pole itself, where the co-latitude is 90°, half of that is 45°, and sec² 45° equals 2. But more importantly, as you move away from the Pole, the co-latitude gets smaller, and the scale expansion grows. Look at Figure 23.2 and you'll see this in action. The meridians are straight lines, the parallels are concentric circles, but the spacing between the parallels is increasing as the distance from the Pole increases. That's the visual signature of this projection. One more useful fact: the Equator can be projected, which means a hemisphere or even more can be shown on a single chart. That's why this projection is so practical for polar regions. Now let's formalise the properties. First, the scale. The scale is correct at the Pole — that's where the paper touches the Reduced Earth, at the point of tangency. Everywhere else, it expands as sec² (½ co-latitude). The equation to find the scale at any latitude is: scale at any latitude = scale at Pole × sec² (½ co-latitude). Now, I should tell you — that equation is not examinable in the EASA ATPL exam. But I'm going to work through it with you anyway, because understanding it makes the whole concept stick. Let's substitute a real value. Take latitude 78°. Assume the scale at the Pole is 1 over 1,000,000 — that's a representative fraction, meaning one unit on the chart equals one million units on the ground. If the latitude is 78°, then the co-latitude is 12°. Half the co-latitude is 6°. So the equation becomes: scale at 78° = 1/1,000,000 × sec² (6°). Now, sec² (6°) can be written two ways. First, as sec (6°) × sec (6°) — that's just what squaring means. Or alternatively, since secant is the reciprocal of cosine, we can write it as 1/cos 6° × 1/cos 6°. So the full expression is: scale at 78° = 1/1,000,000 × 1/cos 6° × 1/cos 6°. Let me just check the arithmetic so you see the practical effect. Cos 6° is approximately 0.9945. So 1/cos 6° is about 1.0055. Squared, that's about 1.011. So the scale at 78° is roughly 1/1,000,000 × 1.011, which is about 1/989,000. The point is, even at 78° — quite far from the Pole — the scale has only expanded by about one percent. That's why this projection is so good for polar navigation: the distortion stays small over a huge area. So to pull it all together: the polar stereographic is a true perspective projection, built with the light source at the South Pole and the paper touching at the North Pole. Meridians are straight radiating lines, parallels are concentric circles that spread apart as you leave the Pole. The scale is exact at the Pole and expands as sec² of half the co-latitude. And the formula scale at any latitude = scale at Pole × sec² (½ co-latitude) ties it all together — even though it's not examinable, it's the engine behind everything you'll see on this chart.