Lambert’s Conformal Chart - 1 — Page 340, Lesson 305

Let’s pick this up right where the geometry gets interesting. We’ve already seen the simple conical projection, where a cone sits over the Reduced Earth and the graticule is cast onto it. Now I want to walk you through what Lambert actually changed, and why it matters for chart convergence. First, recall the simple conic. In that projection, the angle of the cone was twice the latitude of the parallel of tangency. That single number — the cone angle — fixed the chart convergence. And the convergence itself was defined as the change of latitude times the sine of the parallel of tangency. So if you flew through a change of latitude of, say, 5 degrees, and the parallel of tangency was at 30 degrees north, the convergence was 5 times the sine of 30, which is 5 times 0.5, giving 2.5 degrees. That’s the whole relationship in the simple conic. Now, here’s the key move Lambert made. When he pushed the cone inside the Reduced Earth — so that it cuts through the sphere rather than just touching it — he did not change the cone angle. You can think of this in two equally valid ways. Either you keep the cone angle constant and reduce the size of the cone, or you just push the same cone downwards a bit. Both descriptions give the same result: the cone angle stays exactly what it was. And because the cone angle hasn’t changed, neither has the ‘n’ factor — that’s the constant of the cone. This is the factor that relates the convergence to the change of longitude. Since the cone angle is unchanged, the ‘n’ factor is unchanged, and therefore the chart convergence remains the same as it was in the simple conic. But here’s the subtlety. In the simple conic, that convergence was originally defined by the parallel of tangency — the latitude where the cone just touched the sphere. In the Lambert chart, the cone no longer touches; it cuts through. So the convergence is now defined by a different reference line: the parallel of origin. That’s the latitude where the cone’s axis of symmetry meets the sphere in a way that defines the projection’s origin. So we end up with a clean separation of two ideas. The scale is correct at the standard parallels — those are the two latitudes where the cone actually cuts through the Reduced Earth, and where the projection is true to scale. But the convergence factor — the ‘n’ factor that governs how meridians converge — is defined by the parallel of origin, not by the standard parallels. That’s the heart of the Lambert conformal chart: scale accuracy lives at the standard parallels, while convergence is locked to the parallel of origin. Keep those two reference latitudes distinct in your mind, because they do different jobs. The standard parallels control how distances are represented; the parallel of origin controls how directions and meridians converge on the chart. Take a look at the figure showing part of a Lambert conical orthomorphic chart — you’ll see how the meridians converge toward the pole and the parallels appear as arcs, all consistent with that fixed convergence factor. Now, one thing to note: this is the first part of the Lambert chapter, and we’re building the foundation. The next step will be how that convergence actually shows up when you measure track angles on the chart, and how you correct for it. But for now, hold onto this: Lambert kept the cone angle, kept the ‘n’ factor, kept the convergence — he just moved the reference from the parallel of tangency to the parallel of origin, while the scale stays true at the standard parallels.