Lambert’s Conformal Chart - 1 — Page 340, Lesson 301

Let’s start with the physical idea behind the whole chart, because everything else hangs off it. Imagine the Earth shrunk down to a small globe — that’s the Reduced Earth. Now picture a cone sitting over it, touching the globe along one circle of latitude. In Figure 21.1, a light source at the centre of the Reduced Earth casts shadows of the graticule — that’s the grid of meridians and parallels — onto the inside surface of the cone. Those shadows are the map. You mark them in, take the cone off, cut it down its slant side, and roll it out flat. What you get is the simple conical projection, and that’s Figure 21.2. Now, the key property of this simple conical projection: scale is correct on the parallel of tangency — the circle of latitude where the cone actually touches the globe. In this example that’s 45°N. That parallel, the one on which scale is correct, is given a special name: the Standard Parallel. And here’s the catch — scale expands on either side of the Standard Parallel. So the further you move north or south away from that parallel, the more the scale stretches. That’s the fundamental distortion of this projection. Look at Figure 21.2 and you’ll see something interesting about longitude. When the cone is flattened out, all 360° of longitude are represented, but not as a full circle — they’re squeezed into a sector of a circle. In this case, with the parallel of tangency at 45°N, the sector is 255°. The size of that sector is controlled by the latitude you chose as the parallel of tangency. The higher the latitude you choose, the larger the sector becomes. So a high-latitude tangency gives you a big arc; a low-latitude one gives you a small arc. Let me make that concrete with Figure 21.3. Take the parallel of tangency at 45°N. If you use that parallel as the base of a triangle formed by the cross-section of the cone, you get an isosceles triangle — two equal sides — and the angle at the apex is 90°. Now change the parallel of tangency to 60°N, and the apex angle becomes 120°. Push it to the ultimate case, 90°N — that’s the North Pole itself — and you get a flat sheet of paper, with the angle at the pole being 180°. So there’s a clean rule here: the angle at the apex of the triangle is always twice the parallel of tangency. 45° gives 90°, 60° gives 120°, 90° gives 180°. That apex angle has another name — it’s called the angle of the cone. And it’s the angle of the cone that determines the arc of the sector formed when you lay the cone flat. Think about the extremes. At a 90° parallel of tangency, you get a 180° cone angle, which gives no missing gap at all — the sector is a full circle. At a high latitude like 60°N, the paper sector forms quite a large arc, so the ‘missing’ sector — the gap you’d have to fill to make a full circle — is small. At 45°N, the sector is 255°, which makes the missing sector 105°. So the higher the parallel of tangency, the bigger the sector and the smaller the gap. Now here’s the mathematical relationship that ties it all together. The sector formed, compared to the original 360° of longitude, is determined by the sine of the parallel of origin. The formula is: arc of sector equals change of longitude times sine of the parallel of origin. Let’s plug in the numbers. Change of longitude is 360°, sine of 45° — and 360° times sine 45° gives you 255°. That’s where the 255° comes from. What we’re really saying is this: 360° of longitude — and note, that’s the change of longitude between the 180°E and the 180°W meridian in Figure 21.2, going the long way round, not across the gap — that full 360° is represented by only 255° of angle of inclination between the meridians as drawn on the simple conic chart. So the meridians, instead of spreading out a full 360°, are squeezed together into a 255° sector. That’s the geometric heart of the simple conical projection, and it’s the foundation you’ll build on when we move into Lambert’s Conformal Chart proper.