The Polar Stereographic Chart — Page 367, Lesson 322

Let's start with the polar stereographic projection. I want to walk you through what makes it special, because of all the projections we've discussed, this is the only one that's a true geometric projection — a true perspective projection. That means it's actually constructed by projecting light rays from a real point, rather than by mathematical transformation alone. Here's the setup, and I want you to picture it clearly. We take a flat surface and place it touching the North Pole. That point where the paper touches the Earth 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 it's exactly on the opposite side of the Earth. The light from that source shines through the Earth's surface and casts shadows of the graticule — the grid of meridians and parallels — onto our flat paper. That's the geometrical projection. Now, let's look at what that graticule actually looks like. The meridians — the lines of longitude — come out as straight lines radiating from the Pole. The parallels — the lines of latitude — come out as concentric circles, all centered on the Pole. But here's the key point: those circles are not evenly spaced. Their spacing from the Pole depends on something called scale expansion from the Pole. The further you get from the Pole, the more the scale stretches. To understand this spacing, we need a new concept: the co-latitude. The co-latitude is simply 90° minus the latitude. So if the latitude is 60°, the co-latitude is 30°. Simple enough. Now let's trace the light paths. 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°. In the triangle formed by that light path, the opposite side and the adjacent side are both the radius of the Earth, so we have a right-angled triangle with a bottom angle of 45°. There's a geometrical proof — which you don't need to understand, but you must remember the result — that the angle the light path makes with the axis of the Earth is always a function of half the co-latitude. And the result is this: the scale expands away from the Pole at a rate of the secant squared of half the co-latitude. So the formula is: scale expansion = sec² (½ co-latitude). Let me unpack that. Secant is the reciprocal of cosine — sec of an angle is 1 divided by the cosine of that angle. So sec² of half the co-latitude means we take the secant of half the co-latitude and square it. That's the rate at which the scale grows as you move away from the Pole. In the graticule, you can see this effect: the meridians are straight lines, the parallels are concentric circles, but the spacing between the parallels increases as the distance from the Pole increases. The Equator can be projected so that a hemisphere — or even more — can be shown on one chart. Now let's analyse the properties of this chart, starting with scale. The scale is correct at the Pole — that's where the paper touches the Reduced Earth, so there's no distortion there. Everywhere else, the scale expands as sec² (½ co-latitude). There's an equation to find the scale at any latitude: scale at any latitude = scale at Pole × sec² (½ co-latitude). Now, this equation is not examinable in the EASA ATPL exam, but I want to work through it with you anyway, because it shows you exactly how the scale behaves. Let's substitute a particular 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 represents 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° equals 1 over 1,000,000, times sec² of 6°. We can write sec² of 6° as sec of 6° times sec of 6°. And since secant is the reciprocal of cosine, that's 1 over cos 6° times 1 over cos 6°. So the full expression is: 1 over 1,000,000, times 1 over cos 6°, times 1 over cos 6°. What this tells you is that at 78° latitude, the scale is larger than at the Pole — the chart is stretched. And the further you go from the Pole, the more that stretch grows, following this secant-squared relationship. So the takeaway: the polar stereographic chart is a true perspective projection, built by projecting light from the South Pole onto a plane touching the North Pole. Meridians are straight radiating lines, parallels are concentric circles with increasing spacing. And the scale — correct only at the Pole — expands as the secant squared of half the co-latitude.