Direction, Latitude and Longitude 1 — Page 20, Lesson 29

I want to walk you through a new topic now — Great Circle Vertices. This is a concept that builds directly on the latitude and longitude system we've been using, and it's important for understanding how a Great Circle route behaves as it curves across the Earth. Let me start by defining what a vertex is. The northern vertex of a Great Circle is simply the most northerly point on that Great Circle. Similarly, the southern vertex is the most southerly point on that same Great Circle. So if you imagine a Great Circle — the shortest path between two points on a sphere — it will have one point where it reaches its highest latitude in the northern hemisphere, and one point where it reaches its lowest latitude in the southern hemisphere. Those are the vertices. Now, here's an important relationship: the vertices are antipodal. That means they are directly opposite each other on the Earth. The Great Circle distance between them is exactly 10,800 nautical miles. That's half the circumference of the Earth along a Great Circle, which makes sense because antipodal points are 180 degrees of arc apart. Let me give you the exact geometric relationship. The vertices lie on a meridian and its anti-meridian — that is, a line of longitude and the line of longitude exactly 180 degrees opposite it. And the latitude values of the two vertices are equal in magnitude but opposite in sign. For example, if the southern vertex of a Great Circle is at 63 degrees South, 170 degrees West, then its northern vertex will be at 63 degrees North, 010 degrees East. Notice the latitude is the same number — 63 — but one is South and the other is North. And the longitudes are 170 West and 010 East, which are 180 degrees apart. Now, at either vertex, something very specific happens to the direction of the Great Circle. At the vertex, the direction of the Great Circle will be either East — that's 090 degrees True — or West — that's 270 degrees True. So the Great Circle is running exactly east-west at its most northerly or most southerly point. Knowing the coordinates of either vertex is very useful because it allows us to calculate where and at what angle the Great Circle crosses the Equator. Here's the rule: a Great Circle will cross the Equator at two points whose longitude is 90 degrees of longitude change away from either of its vertices. So using the example I just gave, if the vertex is at 170 West, then the Great Circle would cross the Equator at 080 degrees West and 100 degrees East. That's 90 degrees west of 170 West, and 90 degrees east of 170 West. The track angle at which the Great Circle crosses the Equator is based on the latitude of each vertex. The track angle depends on which direction the Great Circle is travelling from the vertex — whether it's going East or West. Let me walk through the example from the book. Take that same Great Circle with a vertex latitude of 63 degrees. If you are travelling Eastbound from the northern vertex, the track angle at the first Equator crossing would be 90 degrees plus 63 degrees, which equals 153 degrees. Then, having passed the southern vertex and still travelling East, the track angle at the second crossing of the Equator would be 90 degrees minus 63 degrees, which equals 027 degrees. If instead the original direction had been West from the northern vertex, the track angles crossing the Equator would be the reciprocals — that is, 207 degrees and 333 degrees. So the key takeaway here is that the vertex latitude gives you the angle at which the Great Circle crosses the Equator, and the direction of travel — eastbound or westbound — determines whether you add or subtract that angle from 90 degrees to get the actual track angle.