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

All right, let’s get into the Lambert’s Conformal Chart. This is a big one in General Navigation, so I want to build it up properly from the ground. First, the headline: the Lambert projection is a non-perspective chart. Let me unpack that. A perspective chart is one you could, in theory, create by placing a light source inside a transparent Earth and projecting the graticule onto a flat surface. The Lambert chart is not made that way. Instead, we take a cone, bring it inside the Reduced Earth, and then we have to make some mathematical adjustments to the projection so that the chart becomes orthomorphic. So the whole thing is built by mathematics, not by geometry or projection from a point. Now, what does orthomorphic mean? It means the chart preserves shape — angles on the chart are the same as angles on the Earth. That’s the property we’re engineering for, and on a Lambert chart it’s achieved by mathematical construction. That’s a key phrase: the chart is orthomorphic by mathematical construction. Let’s look at the properties in detail, starting with scale. Scale is not constant across a Lambert chart. It is least on the parallel of origin. I’ll come back to what that parallel is in a moment, but for now: scale is smallest there. As you move away from the parallel of origin, the scale expands — it grows — until it becomes correct on the standard parallels. And scale is greatest on the top and bottom parallels of the projection. So you have a scale that’s least in the middle, correct at the two standard parallels, and greatest at the edges. Next, orthomorphism again, but now let’s see how the graticule achieves it. On a Lambert chart, the meridians converge — they come together. But the parallels curve. When you put those two together — converging straight meridians and curved parallels — they cut each other at right angles. That right-angle crossing is what gives you the orthomorphic property. That’s the mechanism. Now let’s look at the graticule itself. The meridians are straight lines radiating from the pole. The parallels of latitude are arcs of concentric circles, and all of those circles are centred at the pole. Here’s the practical point: the pole is usually off the map sheet you’re using. So you don’t see the pole on the chart — it’s somewhere off the edge. The map sheet you actually use is shown as a red broken rectangle on the figure. Now, the parallel of origin. This is the mathematical basis of the projection. It defines the chart convergence. It sits half-way between the two standard parallels. And here’s a very important definition: the sine of the parallel of origin is called ‘the constant of the cone’, and it’s denoted by the symbol ‘n’. So if the parallel of origin is at latitude φ, then n = sin φ. That constant n is the key to the whole projection. Now, chart convergence. This is where the Lambert chart differs from the Earth. On the Earth, the convergence between two given meridians changes with latitude — it’s greatest at the poles and zero at the equator. 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. All the angles are the same. That’s a crucial simplification for navigation. And that gives us the formula for calculating chart convergence: Chart Convergence = change in longitude × sine of the parallel of origin. So if you know the change in longitude between two meridians, and you know the sine of the parallel of origin — which is your constant of the cone, n — you multiply them together and you get the chart convergence. That’s the angle between the two meridians as drawn on the chart. Let me tie it all together. The Lambert chart is a non-perspective, mathematically constructed projection. It’s orthomorphic because the converging straight meridians and the curved parallels cut at right angles. Scale is least on the parallel of origin, correct on the standard parallels, and greatest at the top and bottom parallels. The parallel of origin sits half-way between the standard parallels, and its sine is the constant of the cone, n. And because the meridians are straight lines, convergence doesn’t vary with latitude — it’s simply change in longitude times sine of the parallel of origin. That’s the foundation. Once you’ve got the constant of the cone and the convergence formula, the rest of Lambert chart work — great circle plotting, track angles, all of it — builds directly on these.