Mercator Charts - Properties — Page 304, Lesson 271

I want to walk you through the properties of Mercator charts, starting with how rhumb lines and great circles behave on them. Let’s look at Figure 18.6, which shows a Mercator chart with rhumb lines and great circles. On a Mercator chart, a rhumb line — that’s a line of constant true direction — appears as a straight line. A great circle, which is the shortest path between two points on the Earth’s surface, generally appears as a curved line. Now, apply these rules to the chart in Figure 18.7. Imagine a round-the-world trip from London to Los Angeles, USA, then to Auckland, New Zealand, then to Singapore, and finally returning to London. All the tracks drawn on that chart are rhumb line tracks. From London to LA, the rhumb line track is approximately 257° True. From LA to Auckland, the rhumb line track is approximately 221° True. From Auckland to London, the rhumb line track is approximately 301° True. Notice that the Auckland-to-London rhumb line passes through Singapore. Now, if you draw in the great circle tracks for the same route, you get Figure 18.8. Look at the shapes of those great circles. The Equator and the meridians — which are themselves great circles — project as straight lines on a Mercator chart. All other great circles project as curved lines that are concave to the Equator, which means they are convex to the nearer Pole. Let’s move into the mathematical calculations. Given the 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. The formula is: Conversion Angle (CA) = ½ Earth Convergency (EC) Or, more practically: Conversion Angle (CA) = ½ × change of longitude × sine of the mean latitude Let me define those terms. Change of longitude, often written as ch.long, is the difference in longitude between the two points. Mean latitude is the average of the two latitudes. So the conversion angle is half the product of the change of longitude and the sine of the mean latitude. Now review Figure 18.8 and consider the route from London to Los Angeles. The rhumb line direction is approximately 257° True. The conversion angle between the rhumb line and the great circle is calculated using the formula. Using approximate values: change of longitude is 120°, mean latitude is 45°. So: CA = ½ × 120 × sin 45° CA = 60 × 0.707 CA = approximately 42° Therefore, the great circle track from London to LA, measured at London, is approximately 257° plus 42°, which equals 299° True. Now calculate the great circle track direction from LA to London, measured at LA. The rhumb line track from LA to London is the reciprocal of 257° True — that’s 077° True. The conversion angle is still approximately 42°. So the great circle track from LA to London, measured at LA, is approximately 077° minus 42°, which gives 035° True. You could carry out similar exercises for all the other sectors. However, a problem occurs on sectors that cross the Equator. For example, on the Los Angeles to Auckland sector, you might argue that the conversion angle is zero because the mean latitude is 0° — the Equator. In those situations, you have to divide the sector into two parts. The first sector from LA to the Equator has a mean latitude of approximately 18°, and the second sector from the Equator to Auckland also has a mean latitude of approximately 18°. You calculate the conversion angle separately for each half. There are two exceptions to the rule that great circles curve on a Mercator chart. The Equator and the meridians are straight lines — because they are also rhumb lines. So those great circles project as straight lines.