Lambert’s Conformal Chart - 2 — Page 349, Lesson 312

Let’s pick up right where we left off — we’ve built the Lambert chart, we know it’s conformal, we know it’s a cone laid flat. Now I want to walk you through the two big ideas that make this chart tick: earth convergence and chart convergence, and then we’ll see why a great circle curves the way it does on the chart. First, earth convergence — also called convergency. This is the difference in inclination between two meridians on the Earth. Think of two meridians, say 10°W and 20°W. On the globe they start parallel at the equator, but they lean toward each other and meet at the pole. The angle between them at any given latitude — that’s earth convergence. Another way to say it: it’s the change in direction of a great circle track between two meridians on the Earth. So if you fly a great circle from one meridian to the next, your heading changes by exactly the earth convergence between those two meridians. Now chart convergence. Same idea, but on the chart. It’s the difference in inclination between two meridians on the chart — the angle between the two meridian lines as drawn on the flat paper. And equivalently, it’s the change in direction of a straight line track between two meridians on the chart. So on the chart, if you draw a straight line crossing two meridians, the angle it makes with each meridian differs by the chart convergence. Here’s the key contrast: on the real Earth, meridians converge to the pole. On the Lambert chart, the meridians are straight lines that converge to a point — the apex of the cone. But the amount they converge on the chart is not the same as the amount they converge on the Earth. That difference is the whole story of this chart. Now let’s bring in the conversion angle. This is the difference between a rhumb line and a great circle. And here’s the formula you need to remember: conversion angle = ½ ch.long × sin mean lat. Let me unpack that. ch.long is the change of longitude between the two meridians you’re considering. mean lat is the mean latitude — the average of the two latitudes you’re working between. And the sine of that mean latitude, times half the change of longitude, gives you the conversion angle. So conversion angle is half the earth convergence. That’s the relationship — the conversion angle is exactly half of the earth convergence between those two meridians. Now, on the chart, we have a parallel idea. The difference between a rhumb line and a straight line on the chart is half the chart convergence. And the formula for that is: difference between rhumb line and straight line = ½ ch.long × sin parallel of origin. Notice the difference — on the Earth we use the sine of the mean latitude; on the chart we use the sine of the parallel of origin. The parallel of origin is the latitude where the cone touches the Earth — the reference parallel, the one that defines the chart’s convergence. So on the chart, the rhumb line and the straight line differ by half the chart convergence, and that’s computed using the parallel of origin, not the mean latitude. Now here’s the beautiful part — this difference between chart convergence and earth convergence is exactly why great circles are concave to the parallel of origin on the Lambert projection. Let me walk you through the reasoning. Imagine a Lambert chart. Take two meridians, say 20° apart in longitude. Now consider three different latitudes: 47°N, 45°N — and 45°N is our parallel of origin — and 43°N. So we have three horizontal lines on the chart, one at each latitude, spanning the same 20° of longitude. Now draw straight lines joining the meridians at each of these latitudes. So you get three straight lines, one at 47°N, one at 45°N, one at 43°N, each crossing the same two meridians. Now consider the rhumb line in each case. The rhumb line crosses each meridian at a constant angle. The difference between the rhumb line and the straight line — that’s half the chart convergence, and it equals ½ ch.long × sin parallel of origin. Let’s plug in the numbers: ½ × 20° × sin 45°. The sine of 45° is 0.7071. So ½ × 20° × 0.7071 = 7.071°. And here’s the punchline — that’s the same in all three cases! Because the parallel of origin is fixed at 45°N, the half chart convergence is identical at 47°N, 45°N, and 43°N. All three rhumb lines differ from their straight lines by the same 7.071°. Now let’s calculate the angle that the great circle makes with the rhumb line. That’s the conversion angle, and it’s ½ ch.long × sin mean latitude — for each line. And here’s where it gets interesting: calculating this value gives us three different answers, one for each latitude. Because the mean latitude is different for each line — at 47°N the mean latitude is higher, at 43°N it’s lower — so the sine of the mean latitude differs, and therefore the conversion angle differs. So let’s put it together. At each latitude, the rhumb line sits at a fixed angle relative to the straight line — the same 7.071° everywhere. But the great circle sits at a different angle relative to the rhumb line at each latitude — the conversion angle changes with latitude. So the great circle doesn’t stay at a constant angle to the straight line; it bends. And because the conversion angle is larger at higher latitudes and smaller at lower latitudes, the great circle curves — and it curves so that it’s concave toward the parallel of origin. That’s the geometric reason: the chart convergence is constant, but the earth convergence varies with latitude, and that mismatch is what makes the great circle bow toward the parallel of origin on the Lambert chart. So the takeaway: on the Lambert chart, a great circle is not a straight line — it’s a curve that bends toward the parallel of origin. The straight line on the chart is the great circle only at the parallel of origin itself. Everywhere else, the great circle curves, and the amount of curvature is governed by the difference between chart convergence and earth convergence. That’s the core of Lambert’s Conformal Chart — the convergence story. Let’s keep going.