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

I want to walk you through the start of Lambert’s Conformal Chart. We’re going to begin by understanding why we even need conical charts in the first place, then look at the background to the Lambert projection, and finally start on the simple conic projection itself. Let’s start with the limitations of the Mercator chart. You’ve already learned that the Mercator chart has some very powerful properties, but it has two key limitations. First, Great Circles are not projected as straight lines on a Mercator chart. Second, the chart is not constant scale — in fact, scale changes quite rapidly as you move north or south in latitude. For about the first 400 years of the Mercator chart’s life, navigators were happy to follow Rhumb Line tracks — that is, tracks of constant direction — because they were steering by compass. A Rhumb Line is a line that crosses all meridians at the same angle, so if you steer a constant compass heading, you follow a Rhumb Line. That worked fine when you had to steer by magnetic compass. But that situation started to change from about 1960 onwards, when automatic computing became available. With computers, it became possible to calculate a desired Great Circle track direction using spherical trigonometry formulae. The computers built into INS — Inertial Navigation System — IRS — Inertial Reference System — FMS — Flight Management System — and GPS — Global Positioning System — all do this as a matter of course. 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, on a Mercator chart, the aircraft will appear to go off track in the middle of the leg, then recover back to track again — because the Great Circle curves on that projection. There’s another advantage too. 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, being able to use a ruler is very useful — most aircraft do not have a large plotting table like a ship does. So that brings us to the background of the Lambert projection. The basis of the Lambert projection is the simple conical projection. However, the simple conical projection has some limitations — not least that it is not orthomorphic. Let me define that term: orthomorphic means shape-preserving. On an orthomorphic chart, angles measured on the chart are equal to the corresponding angles on the Earth’s surface. The Mercator chart is orthomorphic; the simple conical projection is not. A Swiss mathematician named Johann Heinrich Lambert made some modifications to the basic conic in 1777. The result is the orthomorphic, near constant scale chart we call Lambert’s Conformal Conic Chart. The word “conformal” here means the same thing as orthomorphic — it preserves angles and shapes locally. The approach we’ll take in this chapter is first to derive some of the properties of the simple conic projection. Some of these properties, 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. Finally, we’ll finish up with the properties of the Lambert projection itself. Now let’s start with the simple conic projection and its convergence. It is possible to project the graticule of the Earth — that is, the grid of parallels of latitude and meridians of longitude — onto the inside surface of a cone. This process gives us the conical family of projections, from which Lambert’s is derived. Here’s how it works: a cone is placed over a Reduced Earth — that’s a small-scale 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. You can see this illustrated in Figure 21.1. So to summarise what we’ve covered: we need conical charts because the Mercator chart doesn’t show Great Circles as straight lines and its scale changes rapidly. Lambert modified the simple conic in 1777 to make it orthomorphic and near constant scale. And the simple conic projection starts by placing a cone tangent to the Reduced Earth along a parallel of latitude, with its apex on the Earth’s axis.