Lambert’s Conformal Chart - 1 — Page 340, Lesson 298

Let’s start with the why, because that’s what drives everything else in this chapter. We’ve already seen the Mercator chart and its powerful properties — but it has two real limitations. First, great circles are not projected as straight lines on a Mercator. Second, the chart is not constant scale — in fact, scale changes quite rapidly as you move in latitude. Now, why did navigators put up with that for so long? Because for the first 400 years or so of the Mercator’s life, they were steering by compass. That means they needed a constant track direction — a rhumb line — and the Mercator gives you exactly that. So the trade-off was acceptable. But that situation started to change from about 1960 onwards, when automatic computing became available. With spherical trigonometry formulae, it became possible to calculate a desired great circle track direction, and the computers built into INS, IRS, FMS and GPS do this as a matter of course. So now, if the aircraft is going to be steered along a great circle, it would be very helpful to have a chart on which a great circle is a straight line. Otherwise, the aircraft will appear to go off track in the middle of the leg, then recover back to track again — which is confusing and operationally messy. There’s also the scale issue. If the scale were to remain constant on a chart, we could measure distances with a ruler instead of having to use a pair of dividers and open them to different distances at different latitudes. For aviation that matters, because most aircraft do not have a large plotting table like a ship does — so being able to use a ruler is genuinely useful. That brings us to the background of the Lambert projection. The basis of the Lambert projection is the simple conical projection. But the simple conical projection has limitations — not least that it is not orthomorphic. Orthomorphic means shape-preserving — angles on the chart match angles on the ground. A Swiss mathematician named Johann Heinrich Lambert made some modifications to the basic conic in 1777. The result is the orthomorphic, near-constant-scale Lambert’s Conformal Conic Chart. Now, the approach I want to take in this chapter is to first derive some of the properties of the simple conic projection. Some of these — particularly the chart convergence — are also found in the Lambert projection. Then we’ll examine why Lambert made his changes and what effect they had. And we’ll finish up with the properties of the Lambert projection itself. So let’s look at the simple conic projection and its convergence. It is possible to project the graticule of the Earth onto the inside surface of a cone. That process gives us the conical family of projections, from which Lambert’s is derived. Here’s the setup: a cone is placed over a Reduced Earth — that’s a small model of the Earth — in such a way that the cone is tangential with the Reduced Earth along a parallel of latitude. The apex of the cone will lie on the extended line of the Earth’s axis, as shown in Figure 21.1. So the key idea is that the cone touches the Earth along one parallel — that’s the parallel of tangency — and the cone’s apex sits on the extended Earth’s axis. That geometry is the foundation for everything that follows, because the way the graticule gets projected onto that cone determines the convergence properties we’ll be deriving next.