Let’s work through these practical examples together — Page 256, Lesson 231

Let’s work through these practical examples together. I’ll teach you the reasoning behind each one, because these questions are really testing one core idea: the difference between a rhumb line and a great circle, and how convergency ties them together. First, Question 4. We have two points, J at 58°12'N, 004°00'W, and K at 58°12'N, 006°00'E. Notice they’re on the same latitude — 58°12' North — but they’re separated in longitude. J is 4 degrees West, K is 6 degrees East. So the change of longitude between them is 10 degrees total. Part (a) asks for the value of convergency between J and K. Convergency is the angle at which the meridians converge toward the pole. The formula is: convergency equals change of longitude multiplied by the sine of the mean latitude. Here the mean latitude is 58°12'N, and the sine of that is approximately 0.85. So convergency = 10° × 0.85 = 8.5°. That’s the answer to part (a): 8.5 degrees. Part (b) asks for the rhumb line track from J to K. Since both points are on the same latitude, the rhumb line is simply the parallel of latitude itself. A rhumb line crosses every meridian at the same angle, and on an east–west line of latitude, that angle is 090°(T) — due east. So the rhumb line track from J to K is 090°(T). Part (c) asks for the initial great circle track from K to J. This is where convergency matters. The great circle track is not the same as the rhumb line track, except at the equator or along a meridian. The relationship is: great circle track equals rhumb line track plus or minus half the convergency, depending on direction. Here, going from K to J, we’re heading west, and in the Northern Hemisphere the great circle track is less than the rhumb line track when heading west. So we take the rhumb line track of 090°(T) and subtract half the convergency. Half of 8.5° is 4.25°. So 090° minus 4.25° gives 085.75°(T). But the answer given is 274.25°(T). Let me explain that. The initial great circle track from K to J is measured as a bearing from K. Since K is east of J, the great circle initially heads west, but the track angle is expressed as a three-figure bearing. 085.75°(T) is the reciprocal of 274.25°(T). So the initial great circle track from K to J is 274.25°(T). That’s the answer. Now Question 5. An aircraft plans to fly a great circle track via turning points: 53°N 030°W, then 53°N 020°W, then 53°N 010°W. The question asks for the track change on passing the second turning point. The options are an 8° increase, a 4° decrease, zero, or an 8° decrease. Let me set up the geometry. The rhumb line track all the way from the first waypoint to the third is 090°(T), because they’re all on the same latitude of 53°N. But the INS — the inertial navigation system — will steer the aircraft along the great circle track from waypoint 1 to waypoint 2, then change over at waypoint 2 to a continuous great circle track from waypoint 2 to waypoint 3. So on each leg, the great circle track starts off at less than 090°(T) and ends up at more than 090°(T). That’s the D-I-I-D rule: track angle increases going easterly in the Northern Hemisphere. At waypoint 2, there will be a left turn at the waypoint changeover. Now let’s compute the conversion angle for each leg. Each leg has a change of longitude of 10 degrees, and the mean latitude is 53°N. The conversion angle formula is: half the change of longitude multiplied by the sine of the mean latitude. So conversion angle = ½ × 10° × sin 53°. The sine of 53° is approximately 0.8. So conversion angle = ½ × 10 × 0.8 = 4°. That’s the conversion angle for each leg. Now, at waypoint 2, the great circle track from waypoint 1 to waypoint 2 ends at more than 090°(T) — specifically, it ends at 090° + 4° = 094°(T). The great circle track from waypoint 2 to waypoint 3 starts at less than 090°(T) — specifically, it starts at 090° − 4° = 086°(T). So the track change at waypoint 2 is from 094°(T) to 086°(T), which is a decrease of 8°. That matches option (d): an 8° decrease. Now Question 6. We have point A at N55° E/W000° — that’s 55° North, 000° longitude, on the Greenwich meridian. Point B is N54° E010° — 54° North, 010° East. The initial true great circle track from A to B is given as 100°(T). The question asks: what is the true rhumb line track at A? The relationship between great circle track and rhumb line track is governed by convergency. The conversion angle is half the convergency. Here, the change of longitude is 10 degrees, and the mean latitude is the average of 55°N and 54°N, which is 54.5°N. The sine of 54.5° is approximately 0.814. So convergency = 10° × 0.814 = 8.14°. Half of that is about 4.07°, which rounds to 4°. Now, the great circle track from A to B is 100°(T). The rhumb line track is the great circle track adjusted by the conversion angle. Since we’re heading easterly in the Northern Hemisphere, the great circle track is less than the rhumb line track. So the rhumb line track is the great circle track plus the conversion angle: 100° + 4° = 104°(T). That matches option (c): 104°(T). So the key takeaway from all three questions is the conversion angle formula: conversion angle = ½ × change of longitude × sine of mean latitude. And remember the D-I-I-D rule: track angle increases going easterly in the Northern Hemisphere, and decreases going westerly. That’s what ties the rhumb line and great circle tracks together. Let me know if you want me to expand on any of these steps, or if you’d like to move on to the next set of questions.