Lambert’s Conformal Chart - 1 — Page 349, Lesson 307

Right, let's get into the Lambert’s Conformal Chart. We've already set up the projection, so now I want to look at the two big consequences of how it's built: what happens to Rhumb Lines, and what happens to Great Circles. First, let's pin down the key fact about convergence. Because the meridians on a Lambert chart are straight lines, the chart convergence between any two selected meridians will not change with latitude. In other words, chart convergence is constant on a Lambert chart. Look at Figure 21.14 in your mind: we take the meridians through points A, C and E, and we parallel them through B, D and F respectively. That forms angles marked 'CC', and they're all equal to each other. So no matter where you are north or south, the angle between two meridians on the chart stays the same. Now, Rhumb Lines. Except for the meridians, which appear as straight lines, Rhumb Lines are curves concave to the pole of the projection. And the parallels of latitude are Rhumb Lines too — so they're also curves concave to the pole. So a Rhumb Line bends away from the pole, curving in towards it. Now Great Circles. Except for the meridians, which again appear as straight lines, Great Circles appear as curves concave to the parallel of origin. And here's the key practical point: a straight line in an east-west direction most nearly represents a Great Circle when drawn between two positions on the parallel of origin. Let me walk you through why, with the series of diagrams. Imagine a spherical Earth — the real Earth, or a Reduced Earth. The meridians converge towards each other with increasing latitude. Now imagine three Great Circle tracks at different latitudes. That's Figure 21.15 — Great circles on real Earth. Now, what happens when we project this real Earth situation onto a Lambert chart with a parallel of origin of 45°N? The Earth's meridians are straightened out into straight-line chart meridians. That means the formerly straight-line Great Circles are stretched outwards at latitudes higher or lower than the parallel of origin. So the Great Circles are straight lines at the parallel of origin, and curves concave to the parallel of origin at any other latitude. That's Figure 21.17. But — and this is important for your plotting — the amount of curvature from the straight line is exaggerated in Figure 21.17 simply to make the explanation clearer. In fact, there is very little curvature compared with a straight line. For all practical purposes, including plotting, Great Circles on a Lambert chart may be treated as straight lines. Now, one contrast to keep in your head: the amount of curvature on a Rhumb Line is far greater than on a Great Circle. We'll come back to that point in the next chapter. And there's one more relationship I want to make sure you've got, and it's the headline of Figure 21.16: Chart Convergence equals Earth Convergence at the Parallel of Origin. That's the mathematical basis of the whole projection — the parallel of origin defines the chart convergence. So when you're working out convergence on a Lambert chart, you're using the convergence that exists on the real Earth at that one special parallel. So to sum up where we are: chart convergence is constant; Rhumb Lines are curves concave to the pole; Great Circles are curves concave to the parallel of origin, but so nearly straight that we treat them as straight lines for plotting; and the whole thing is anchored by the parallel of origin, where chart convergence equals Earth convergence.