NavList:

David, you wrote:
"What could you use the formula for except for passing exams?  Well I suppose it might help you to know what cold weather gear to take with you, or if you were likely to encounter pack ice. Maybe it could be used in broadcasting or to predict satellite orbits."

Yes, it's for exams.

There are many better practical ways of managing this. The usual textbook advice suggests that you might want the highest latitude of the great circle because you want to stay below the latitude of icebergs. This is rotten navigational advice. First of all, it puts the idea in the young navigator's head that such a thing exists. And an idea learned early tends to be "sticky" which is what makes it dangerous. There's no magic latitude for ice in the North Atlantic or anywhere else. There are regions that can be plotted where ice is more dangerous and where icebergs will be found in certain months of the year. They are not latitude limits. And in the modern world, we have detailed data on ice available en route on any day of a voyage. So you plot those regions on your chart, and you stay away.

You can get the great circle "vertex" (an excellent example of forgettable and pointless jargon) without calculation using a gnomonic chart. You draw your straight line, and you can directly see the highest latitude. You can read it off to the nearest tenth of a degree easily. You can also see directly that the longitude of that point with highest latitude is ill-defined which a pure trigonometric calculation will not reveal. Unfortunately, gnomonic charts are rather rare on paper. Fortunately there are easy digital solutions.

Most digital mapping software can plot a great circle track in a flash. There's nothing to it. 

You can also get the great circle vertex by pulling a slightly elastic, low friction string across a globe. This is really the best way to test out great circle courses because it can immediately be extended to exclude certain regions just by blocking them off with thumbtacks (easier with a digital globe, but the principle is 100% valid with a traditional globe). The purpose of a great circle is to minimize path length. A taut string will do that and can include any changes of geometry as desired. 

In aviation, commercial flights are prohibited from flying over Antarctica despite the fact that some useful great circle routes would pass over the continent. For example, if you're flying from Perth, Australia to Buenos Aires, Argentina, the great circle over Antarctica would be shortest, but it's off-limits. In practice some of those flights stop mid-Pacific, maybe at Fiji or Tahiti (I'm just guessing --I haven't checked to see where actual flights stop). But we can easily stretch a string between the two cities and then push it out of the Antarctic exclusion zone to define the parameters for a non-stop flight. The result will be the shortest path.

As we discussed last year, the advice found in some books that you should take the direct great circle between two cities and then "cut out" the section above some fixed "highest latitude" and replace that with a small circle leg is just wrong. Anytime a course has a sharp turn in it in the middle of nowhere (unless you're avoiding a threat that ends in a single point), then you know that something is wrong. A string pulled taut, even if you just imagine its appearance in your head, always yields the shortest path.

David, you mentioned latitudes for satellites as a possible application. The max latitude for a satellite is usually known immediately because that's just the orbital inclination. And this equivalence provides yet another method for finding the maximum latitude on an ocean track great circle: if the track crosses the equator, the course bearing at that time relative to east/west is identically equal to the maximum latitude on the great circle later. Of course that implies that you've passed through 90° of longitude between the equator crossing and the maximum latitude which is unlikely in any practical case. But it does provide another way to visualize the maximum latitude.

The whole obsession with great circles is mostly motivated by exam questions. These exams and the chapters in textbooks devoted to passing exams have screwed up generations of navigators. How many navigators a thousand miles out in the Pacific have become confused and unable to act because they believe that they need a great circle track to get home to San Francisco? How many navigators in the Atlantic have mistakenly believed that they need to plot a great circle to get from New York to Bermuda? 

Frank Reed