Let me explain what each part means — Page 340, Lesson 303

I want to walk you through the core relationship that governs how meridians appear on a conical projection chart. We start with a formula: angle of inclination of meridians = change of longitude × sine of parallel of origin Now, the angle of inclination of the meridians is what we call chart convergence. So we can rewrite that formula as: chart convergence = change of longitude × sine of parallel of origin This is a general relationship that applies to this type of projection. Let me explain what each part means. Change of longitude — that's the difference in longitude between two meridians, measured in degrees. For example, the meridians at 10°E and 30°E have a change of longitude of 20°. Parallel of origin — that's the latitude line where the cone of the projection touches or is defined relative to the Reduced Earth. On a simple conic projection, this is the parallel of tangency. On Lambert's projection, it's a specific reference parallel. Sine of the parallel of origin — you take the latitude of that parallel and find its sine. For a parallel of origin at 45°N, sin 45° is approximately 0.7071. So if you have a change of longitude of 100° and a parallel of origin at 45°N, the chart convergence — the angle between those two meridians on the chart — is 100° × sin 45°, which equals 70.71°. That's illustrated in Figure 21.4. Similarly, if the change of longitude is only 10°, with the same parallel of origin at 45°N, the chart convergence is 10° × sin 45°, which equals 7.071°. That's shown in Figure 21.5. Now, this sine of the parallel of origin has a special name. It's called "the constant of the cone", and it's represented by the symbol "n". So you'll often see the formula written as: chart convergence = change of longitude × n where n = sin(parallel of origin). Now let's move to scale on the simple conic projection. On a simple conic projection, the scale is correct only at the parallel of tangency — that's the latitude line where the paper cone actually touches the Reduced Earth. Figure 21.6 shows this correct scale at that one line. However, the scale expands away from the parallel of tangency rather rapidly. As you move north or south from that line, the scale gets larger and larger, introducing significant distortion. Figure 21.7 illustrates this scale expansion. This rapid scale change was a problem, and that's where Lambert's Conical Orthomorphic Projection comes in. Lambert modified the simple conic to reduce this rapid rate of scale change. Instead of having the cone touch the Reduced Earth at a single parallel of tangency, Lambert did something different. He made the cone of his projection go inside the Reduced Earth. Look at Figure 21.8 — part (a) shows the simple conic with the cone touching at one point, and part (b) shows Lambert's modification with the cone cutting through the sphere. The result is that the scale is now correct at two points — two latitude lines where the cone intersects the Reduced Earth. These are called the Standard Parallels. Figure 21.9 shows these two standard parallels where scale is correct. Now here's what happens to scale on Lambert's projection. The scale expands outside the standard parallels — those are the yellow parts in Figure 21.10. And the scale contracts between the standard parallels — that's the yellow part in Figure 21.11. This has the effect of evening out the scale error, making it more of a constant scale chart. The old parallel of tangency from the simple conic is renamed the parallel of origin on Lambert's projection. And the scale is least on this parallel of origin — just as it was on the simple conic. To summarize the scale behaviour: - Outside the standard parallels: scale is greater (more expanded) than on the standard parallels. - Between the standard parallels: scale is less (contracted), being least of all on the parallel of origin. Let me give you a concrete example from Figure 21.12. The standard parallels are 43°N and 47°N, and the parallel of origin is 45°N. The upper standard parallel appears one sixth of the way from the top of the chart, and the lower standard parallel appears one sixth of the way from the bottom of the chart. This is called the 'one sixth rule', and it ensures minimum scale variation over the sheet. By placing the standard parallels one sixth from each edge, the scale error is distributed as evenly as possible across the chart.