Mercator Charts - Properties — Page 304, Lesson 271

Let's look at the Mercator chart and what it does to rhumb lines and great circles. I want you to picture the round-the-world route they've drawn: London to Los Angeles, then to Auckland, then to Singapore, and back to London. On a Mercator chart, every one of those tracks is drawn as a straight line, and a straight line on a Mercator chart is a rhumb line — a line of constant true bearing. Let me give you the actual numbers from the chart. From London to LA, the rhumb line track is approximately 257°(T). From LA to Auckland, it's approximately 221°(T). And from Auckland to London, the rhumb line track is approximately 301°(T). Now here's a neat observation: that Auckland-to-London rhumb line passes right through Singapore. That's a nice check on how the chart is laid out. Now, the whole point of this section is what happens when you draw the great circle tracks for the same route. Here's the key rule about shapes on a Mercator chart: the Equator and the meridians are themselves great circles, and they project as straight lines. But every other great circle projects as a curved line — and it's curved in a specific way. It's concave to the Equator, which means it's convex to the nearer Pole. So a great circle bulges toward the pole, away from the Equator. That's the shape rule you need to remember. Now let's get into the mathematics, because given a rhumb line direction, you should be able to calculate the direction of the great circle. The angle between the great circle and the rhumb line is called the conversion angle, abbreviated CA. And here's the formula: Conversion Angle (CA) = ½ Earth Convergency (EC). Or, written out in a more usable form: Conversion Angle (CA) = ½ × ch.long × sin (mean lat). Let me unpack that. ch.long is the change of longitude between the two places — the difference in their longitudes. mean lat is the mean latitude, the average of the two latitudes. And sin is the sine function. So you take half the change of longitude, multiply it by the sine of the mean latitude, and that gives you the conversion angle in degrees. Let's apply this to the London-to-LA leg. The rhumb line direction is 257°(T). We need the conversion angle. Using approximate values: CA = ½ × 120 × sin 45°. The change of longitude is about 120 degrees, and the mean latitude is about 45 degrees. So half of 120 is 60, and sin 45° is 0.707. Multiply 60 by 0.707, and you get approximately 42°. So the conversion angle is about 42 degrees. Now, to get the great circle track measured at London, you add that conversion angle to the rhumb line track: 257° + 42° = approximately 299°(T). So the great circle track from London to LA, measured at London, is about 299°(T). Now let's do the reverse — the great circle track from LA to London, measured at LA. The rhumb line track is the reciprocal of 257°(T), which is 077°(T). The conversion angle is still approximately 42°. So the great circle track is 077° minus 42°, which gives you approximately 035°(T). That's the answer they give. Now, here's a subtle problem that comes up. If you try to do this for a sector that crosses the Equator, you run into trouble. Take the Los Angeles to Auckland sector. The mean latitude is 0° — it's the Equator. And sin of 0° is 0, so the conversion angle would come out as zero. That's clearly wrong, because the great circle is still curved. So in these situations, you have to divide the sector into two parts. First, the sector from LA down to the Equator, where the mean latitude is approximately 18°. Then the second sector from the Equator to Auckland, again with a mean latitude of approximately 18°. You calculate the conversion angle for each half separately. And finally, there are two exceptions to this whole rule about curved great circles. The Equator and the meridians are straight lines on a Mercator chart — because they are also rhumb lines. So those two great circles don't curve; they stay straight. Everything else curves, concave to the Equator, convex to the nearer Pole.