The Polar Stereographic Chart — Page 377, Lesson 331

I want to walk you through the Polar Stereographic Chart, and we're going to pick up right where the geometry of the straight-line track left off. We've already established that the chart convergence between A and B is 120°, because the longitude change is 120° and the 'n' factor is 1. Now let's look at the track angles themselves. If the track angle from B to A is 330°(T) at B, then going back to A, the reciprocal must be 150°(T) going from A to B as it passes over B. In other words, the straight-line track from A to B starts with a track angle of 030°(T) and finishes on a track angle of 150°(T). We've shown from the geometry of the triangle that the track angle increases by 120°. We should expect this — the chart convergence from A to B is 120°, due to the longitude change of 120° and the 'n' factor of 1. Now, there's another way of solving this problem — by use of the Rhumb Line. If latitude A is 75N and latitude B is 75N, then the Rhumb Line track between them is the parallel of latitude of 75N, and therefore must have a Rhumb Line track of 090°(T). The chart convergence between 60W and 60E is 120°. Therefore half chart convergence is 60°. This is the difference between the Rhumb Line and the straight line. So 60° anticlockwise from the Rhumb Line track of 090°(T) gives us an initial straight-line track of 030°(T). The same method would work for finding the track from B to A, measured at B. At B, the Rhumb Line track to A is 270°(T). Sixty degrees clockwise from 270°(T) is 330°(T). Now let's move to Example 2. The question is: at what longitude does the straight-line track from A (70N 40W) to B (70N 80E) on a Polar Stereographic chart reach its highest latitude, i.e. the Great Circle Vertex? The options are 040W, 030W, 020E, and 040E. Start by drawing a diagram of the situation. The highest latitude occurs at the shortest co-latitude distance, i.e. the shortest distance from the Pole. This shortest distance to the Pole must be when the line between the Pole and the track is at right angles to the track. This must occur at mid-meridian — at 020E. So the answer is (c). Now Example 3. On a Polar Stereographic map, a line is drawn from position A (70N 102W) to position B (80N 006E). The point of highest latitude along this line occurs at 035W. What is the initial straight-line track angle from A to B, measured at A? The options are 049°(T), 077°(T), 229°(T), and 023°(T). Again, start by drawing a diagram. This is clearly a new sort of problem, because A and B are at different latitudes. There appears to be no isosceles triangle or right-angled triangle. However, we have not used all the information in the question yet. The point of highest latitude must have some significance, otherwise presumably the examiner would not have mentioned it. So let us draw that in. Now we have a right-angled triangle, between A, the North Pole, and the point of highest latitude. Angle b is the difference between 035W and 102W, so it is 67°. Angle c is 90°. The internal angles of a triangle must add up to 180°. So angle a must be 23°. The answer is (d). Let me make sure you've got the key relationships. The chart convergence is the angular difference between the straight-line track and the Rhumb Line track, and it's governed by the longitude change and the 'n' factor. When the latitudes are equal, the Rhumb Line track is simply the parallel of latitude, and half the chart convergence gives you the angular offset between the Rhumb Line and the straight line. When the latitudes differ, you use the point of highest latitude to construct a right-angled triangle with the Pole, and the internal angles of that triangle give you the initial track angle.