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 conic projection, where the cone sits on the Reduced Earth and touches it along one parallel — the parallel of tangency. Now Lambert takes that same idea and pushes the cone inside the Earth, and that single change is what gives us the Lambert conformal chart. So the first thing I want you to hold onto is this: Lambert did not change the cone angle. He kept the cone angle exactly the same, and instead just moved the cone down a bit, so it cuts through the Earth rather than just touching it. You can think of it either way — keeping the cone angle constant and shrinking the cone, or pushing the same cone downwards. Both descriptions are equally valid, and both give you the same result. Now, why does that matter? Because the cone angle is what fixes the chart convergence. In the simple conic, the angle of the cone was twice the latitude of the parallel of tangency. That relationship set the convergence, and the convergence itself was the change of latitude times the sine of the parallel of tangency. So if you change latitude by a certain amount, the meridians converge by that amount multiplied by the sine of the latitude where the cone touches. Here’s the key point: since Lambert kept the cone angle unchanged, the factor that controls convergence — what we call the ‘n’ factor, or the constant of the cone — also stays unchanged. The chart convergence remains exactly the same as it was in the simple conic. But here’s the subtle shift: in the simple conic, that convergence was defined by the parallel of tangency, the line where the cone touched. In the Lambert chart, because the cone now cuts through the Earth, that defining parallel is no longer the tangency parallel — it’s called the parallel of origin. So we end up with a clean separation of roles. The scale is correct at the standard parallels — those are the two parallels where the cone actually intersects the Earth’s surface. But the convergence factor, the ‘n’ factor, is defined by the parallel of origin, not by the standard parallels. That’s the crucial distinction to remember: scale is tied to the standard parallels, convergence is tied to the parallel of origin. Let me make sure that’s crystal clear, because it’s easy to mix up. The cone angle didn’t change, so the convergence behaviour didn’t change. What changed is only which parallel we use to define that convergence. In the simple conic it was the parallel of tangency; in Lambert it’s the parallel of origin. And the scale being correct at the standard parallels is a separate property — it comes from the cone now intersecting the Earth at two lines instead of touching at one. So when you’re working with a Lambert chart, you’ll always have those two reference parallels in mind: the standard parallels, where the scale is true, and the parallel of origin, which sets the convergence factor. They’re not the same thing, and they serve different purposes. That’s the heart of the Lambert conformal chart’s design.