Lambert’s Conformal Chart - 1 — Page 349, Lesson 306

Let’s start with the big picture. We’ve just brought a cone inside the Reduced Earth, and now we have to make it mathematically work as a chart. The key phrase here is that the Lambert projection is a non-perspective chart. That means it isn’t drawn by simply projecting light rays from a point onto the cone. Instead, we adjust the mathematics to force the chart to be orthomorphic — that’s the technical term for a chart that preserves angles and shapes locally. So the Lambert chart is orthomorphic by mathematical construction, not by geometry. Now let’s look at the scale behaviour, because that’s one of the defining features. On a Lambert chart, scale is least on the parallel of origin. That’s the parallel where the cone touches the Reduced Earth. As you move away from that parallel, the scale expands — it gets larger — until it becomes correct on the standard parallels. Those are the two parallels where the cone cuts through the Earth. And then, beyond them, the scale keeps growing, so it is greatest on the top and bottom parallels of the projection. So the scale profile is: smallest at the parallel of origin, correct at the two standard parallels, and largest at the outer edges of the chart. Next, the graticule — that’s the grid of meridians and parallels. On a Lambert chart, the meridians are straight lines radiating from the pole. They all converge at the pole, like spokes of a wheel. The parallels of latitude are arcs of concentric circles, and all of those circles are centred at the pole. Now, the pole itself is usually off the map sheet you’re using — you don’t see it on the paper. On the figure, the map sheet is shown as a red broken rectangle. So you’re looking at a small piece of a much larger cone. Now, the parallel of origin — this is the mathematical basis of the whole projection. It defines the chart convergence. It sits half-way between the two standard parallels. And here’s a crucial definition: the sine of the parallel of origin is called ‘the constant of the cone’, and it’s denoted by the symbol ‘n’. So when you see ‘n’ in the formulas, that’s what it is — the sine of the parallel of origin. Now let’s talk about chart convergence. On the real Earth, meridians converge — they get closer together as you go toward the poles, and the amount of convergence changes with latitude. But on a Lambert projection, the meridians are straight lines. So, unlike on the Earth, the convergence between two given meridians does not change with latitude. It’s constant across the chart. That’s a key contrast to keep in mind. And that gives us the formula for chart convergence: Chart Convergence = change in longitude × sine of the parallel of origin. In symbols, that’s the change in longitude multiplied by ‘n’, the constant of the cone. So if you know how much longitude you’ve crossed and you know the parallel of origin, you can find the convergence between those two meridians anywhere on the chart — and it will be the same at every latitude. One more thing about the geometry: the meridians converge, but the parallels curve. When you put them together, they cut at right angles. That’s what makes the chart orthomorphic — the graticule lines intersect at 90 degrees everywhere. So, to tie it all together: the Lambert chart is non-perspective, orthomorphic by construction, with scale least at the parallel of origin, correct at the standard parallels, and greatest at the outer parallels. The graticule has straight converging meridians and curved concentric parallels, all centred at the pole. The parallel of origin defines the convergence, and its sine is the constant of the cone, ‘n’. And convergence is simply change in longitude times that constant.