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

We're now moving into the heart of why the Lambert chart behaves the way it does — the relationship between Earth convergence and chart convergence. This is where the projection's quirks become predictable, and it's essential for your navigation calculations. First, let's define our two key terms precisely. Earth convergence, also called convergency, is the difference in inclination between two meridians on the Earth. Think of it as the angle at which two meridians meet as they head toward the pole. It can also be described as the change in direction of a great circle track between two meridians on the Earth. So, as you fly a great circle from one meridian to another, the track direction changes by exactly this amount. Now, chart convergence is the analogous concept, but on the chart itself. It's the difference in inclination between two meridians as drawn on the chart, or equivalently, the change in direction of a straight line track between two meridians on the chart. On a Lambert chart, the meridians are straight lines that converge toward the pole, so they do have a measurable convergence. Here's the critical relationship: the difference between a rhumb line and a great circle is known as the conversion angle, and it is half the earth convergence. The formula is: ca = ½ ch.long × sin mean lat. Let me unpack that. "ca" is conversion angle, "ch.long" is change of longitude, and "sin mean lat" is the sine of the mean latitude — the average latitude between your start and end points. So conversion angle depends on how much longitude you cross and the latitude at which you're operating. Similarly, the difference between a rhumb line and a straight line on the chart is half the chart convergence. The formula for that is: difference between rhumb line and straight line = ½ ch.long × sin parallel of origin. Notice the subtle but crucial difference — instead of "mean lat," we use "parallel of origin." That's the latitude at which the cone of the projection is tangent to the Earth, and it's the mathematical basis of the entire chart. This is why the parallel of origin is so important — it defines the chart's convergence properties. Now, here's where it gets interesting. The difference between chart convergence and earth convergence is what explains 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 — which is our parallel of origin — and 43°N. Draw straight lines joining the meridians at each of these latitudes. These straight lines on the chart represent what a great circle would look like if chart convergence exactly matched earth convergence. Now consider the rhumb line in each case. The difference between the rhumb line and the straight line will be half the chart convergence, and it's equal to ½ ch.long × sin parallel of origin. Let's plug in the numbers: ½ × 20° × 0.7071. The sine of 45° is 0.7071, so that gives us 7.071° — and here's the key point — it's 7.071° in all three cases! Because the parallel of origin is fixed at 45°N, the chart convergence is identical regardless of which latitude you're looking at. The straight lines and rhumb lines maintain a constant angular relationship across the chart. But 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. Here's the catch: calculating this value gives us three different answers, one for each latitude. At 47°N, the mean latitude is higher, so the sine is larger, giving a bigger conversion angle. At 43°N, it's smaller. At 45°N, it exactly matches the chart convergence. So here's the synthesis: the chart convergence is constant — 7.071° for all three latitudes — but the earth convergence varies with latitude. Where they match, at the parallel of origin, the great circle appears as a straight line. Away from the parallel of origin, the great circle curves, and it curves toward the parallel of origin — that's what "concave to the parallel of origin" means. The great circle bends so that its middle is closer to the parallel of origin than its ends. This is the fundamental distortion of the Lambert projection, and understanding it lets you predict how any great circle will appear on your chart. That's the core of this section. The constant chart convergence versus the latitude-dependent earth convergence is what creates the curvature you'll see on every Lambert chart you use.