Lambert’s Conformal Chart - 2 — Page 361, Lesson 318

We are now in the heart of plotting ADF and VOR bearings on a Lambert chart, and I want to walk you through the two big ideas that make this chart so useful in the air. First, let’s talk about how we actually plot a bearing measured at the aircraft. When you’re flying and you get a relative bearing from an NDB, the standard method is to draw a line through the NDB that is parallel to the aircraft’s meridian. The chart convergence has already been automatically allowed for by doing this. This is the method that is always used in plotting. So, if you measure a bearing at the aircraft, you don’t worry about convergence at all—you just draw that parallel line and plot from it. But sometimes you need to plot from the meridian of the NDB itself, and that’s where the convergence comes back in. If you look at Figure 22.12(b), you’ll see that in this case the bearing to plot from the NDB’s meridian is equal to 090° minus chart convergence. Let me give you a concrete example. An aircraft is flying in the Northern hemisphere on a heading of 330°(T), and it measures a bearing of 090°(R) of an NDB. The chart convergence is 3°. First, we find the true great circle bearing of the NDB from the aircraft: 330° plus 090°(R) gives us 060°(T). Now, to plot from a line parallel to the aircraft’s meridian drawn through the NDB, we take the reciprocal, which is 240°. That’s part (a) of the solution. For part (b), we need the bearing to plot from the meridian of the NDB. That is 240° plus the chart convergence of 3°, giving us 243°. The reason we add the convergence here is that the meridians converge toward the pole in the Northern hemisphere, so the bearing measured from the NDB’s own meridian is slightly different from the one measured from the parallel line. Now, here’s a critical rule that you must remember for the exam and for real navigation. When bearings are measured at the aircraft—like NDB/ADF or AWR—it is the aircraft’s compass that is used to add to the relative bearing to obtain the true bearing. So, if your heading is magnetic and you need to correct it to True, you always use the variation at the aircraft. But when bearings are measured at a ground station—like VOR or VDF—you use the variation at the ground station. That distinction is absolutely vital: variation at the aircraft for aircraft-measured bearings, variation at the ground station for ground-station-measured bearings. Now let’s move to the second big idea: what happens when you join two points A and B on a Lambert chart with a straight line. This is where the conversion angle comes in. Let me walk you through Example 1. Two points A and B are plotted, and the straight line joining them measures 250°(T) from the meridian of A. The chart convergence of the meridians through A and B is 6°. We’re in the Northern hemisphere. The key relationship is that the angle between the straight line and the rhumb line is half the chart convergence, which here is 3°. For part (a), the rhumb line track from A to B is the straight line bearing minus half the convergence: 250° minus 3° equals 247°(T). Why do we subtract? Because the straight line on a Lambert chart is a great circle, and in the Northern hemisphere the great circle curves toward the pole, so the rhumb line is slightly less than the straight line bearing. For part (b), we want the approximate great circle bearing of A from B, as given by the straight line, measured from the meridian of B. First, we find the rhumb line bearing of A from B, which is the reciprocal of 247°(T): that’s 247° minus 180°, giving us 067°(T). Then, the straight line bearing is the rhumb line bearing minus half the convergence again: 067° minus 3° equals 064°(T). So the great circle bearing of A from B is 064°(T). Now let’s look at Example 2, which is a bit trickier because it works backwards. The straight line from A to B cuts the mid-meridian in the direction of 300°(T). Measured from the meridian through A, the direction of the straight line is 302°(T). We need three things. For part (a), the rhumb line track from A to B. Here’s the key insight: the rhumb line and the straight line are parallel at the mid-meridian. That’s a fundamental property of the Lambert chart—at the mid-meridian, the great circle and the rhumb line have the same direction. So the rhumb line track from A to B is simply 300°(T). For part (b), we find the chart convergence. The straight line at A measures 302°(T), and the rhumb line is 300°(T). The difference is 2°, and that difference is half the chart convergence. So half the convergence is 2°, meaning the full chart convergence is 4°. For part (c), we want the approximate great circle bearing of A from B, as given by the straight line. First, the rhumb line bearing of A from B is the reciprocal of 300°(T), which is 120°(T). Then we subtract half the convergence, which is 2°, giving us 118°(T). So the approximate great circle bearing of A from B is 118°(T). Let me tie this together. The Lambert chart is conformal, meaning angles are preserved locally, but because the meridians converge, a straight line between two points is a great circle, not a rhumb line. The conversion angle—half the chart convergence—is the correction you apply to convert between the straight line (great circle) and the rhumb line. And when you’re plotting bearings, you either use the parallel line through the point of measurement, which automatically handles convergence, or you apply the convergence correction when measuring from the actual meridian. Take a look at Figure 22.13 to see how the ADF bearing is plotted in that first example. The aircraft is at one point, the NDB at another, and you can see the 060° true bearing and the 240° reciprocal plotted from the parallel line. The chart convergence of 3° is shown, and you can see how it shifts the bearing when measured from the NDB’s meridian. That’s the core of plotting bearings on a Lambert chart. The key takeaways are: always use the parallel line method for aircraft-measured bearings, remember the variation rule for aircraft versus ground station, and apply half the chart convergence when converting between great circle and rhumb line directions.