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

I want to walk you through the Lambert’s Conformal Chart. We’re starting with the principle of a simple conical projection, which is the foundation for understanding how this chart works. Let’s begin with Figure 21.1. Imagine a Reduced Earth — that’s a small, scale model of the Earth — with a light source placed at its very centre. That light source casts shadows of the graticule — the grid of latitude and longitude lines — onto the inside surface of a cone that’s placed over the Earth. Those shadows could be marked in, then the cone is removed, cut down along its slant side, and rolled out flat. That gives you a simple conical projection, as illustrated in Figure 21.2. Now, on this simple conical projection, the scale is correct only on one specific parallel of latitude — the one where the cone just touches the Earth. In the example given, that’s 45°N. This parallel, where the scale is correct, is called the Standard Parallel. On either side of the Standard Parallel, the scale expands — meaning distances get progressively larger than true as you move away from it. Look at Figure 21.2. When the cone is flattened out, the full 360° of longitude around the Earth are represented not as a full circle, but as a sector of a circle — in this case, a sector of 255 degrees. The size of that sector is controlled by the parallel of latitude chosen to be the parallel of tangency. The higher the latitude you choose, the larger the sector will be. Now, let’s move to Figure 21.3. The parallel of tangency here is 45°N. If you take this parallel as the base of a triangle formed by a cross-section of the cone, you get an isosceles triangle. The angle at the apex of that triangle is 90°. If instead we had taken a parallel of tangency of 60°N, the apex angle would be 120°. The ultimate case is a parallel of tangency of 90°N — the North Pole. In that case, the cone becomes a flat sheet of paper, and the apex angle is 180°. Here’s the key relationship: the angle at the apex of the triangle is always twice the parallel of tangency. So for 45°N, apex is 90°; for 60°N, apex is 120°; for 90°N, apex is 180°. That apex angle is also called the angle of the cone. It’s this cone angle that determines the arc of the sector formed when you lay the cone flat. In the ultimate case — a 90° parallel of tangency — you get a 180° cone angle, which gives no missing gap at all; the flattened cone is a full semicircle. If the parallel of tangency is at a high latitude, say 60°N, then the paper sector forms quite a large arc, and the ‘missing’ sector — the gap that isn’t part of the chart — is small. At a parallel of tangency of 45°N, the sector is 255°, making the missing sector 105°. There’s a precise mathematical relationship here. The arc of the sector formed, compared to the original 360° of longitude change, is determined by the sine of the parallel of origin — that’s the parallel of tangency. The formula is: arc of sector = change of longitude × sine of parallel of origin So in our example, with a parallel of origin of 45°N, we have: arc of sector = 360° × sine 45° = 360° × 0.7071 = 255° What this means is that 360° of longitude — which is the change of longitude between the 180°E and 180°W meridian, going the long way around the Earth, not across the gap — is represented by only 255° of angle of inclination between the meridians as drawn on the simple conic chart. That’s the fundamental geometry of how a simple conical projection works.