A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2018 Jan 20, 13:14 -0800
Stan you wrote:
"In Method 1, a great circle route is plotted from the point of departure to the destination and is followed from the point of departure until it crosses the limiting latitude, then the parallel of the limiting latitude is followed until the great circle again crosses it, then the great circle route is followed to the destination. In Method 2, one great circle is plotted from the point of departure tangent to the limiting parallel, and a second from the destination tangent to the limiting parallel. The first is followed from the point of departure to the limiting latitude, then the parallel of the limiting latitude is followed until the second great circle reaches it, then the second great circle is sailed to the destination. Method 1 is easier to plot, but Method 2 results in a shorter course. (In both methods, plotting is done on a gnomonic projection.)"
Method 1 is actually nonsense. It "feels like" navigation, but it's ritual. I strongly suspect that it continues to be popular simply because of some incorrect textbook in the recent history of navigation. The whole point of using great circle segments is to minimize distance travelled. Method 1 emphatically does not do that. It's easiest to see this with an arctic-crossing path. Suppose I need to go from 60°N, 90°W to 60°N, 90°E but I need to avoid latitudes greater than 80°. By Method 1, I would draw a line due north from latitude 60 to 80 along 90 west and then turn along the circle of latitude making a semi-circle along lat 80, running either east or west until I reach 90 east. Then I would run due south. This is obviously much longer than necessary. The easiest way to find the shortest distance path is to get out a globe and a piece of string. Put some pins in the globe (in your imagination, if not in reality) along 80 N to act as a fence for the limiting latitude. Then pull the string between the start and end points. You find great circle segments that "kiss" the parallel of latitude at 80 N, exactly as in method 2. This is also very easy to do graphically on a gnomonic chart: you place a straight edge (or a string!) at the starting location, and then rotate it around until it just touches the limiting latitude of the forbidden zone. Do the same thing from the ending point. Then connect up along the parallel of the limiting latitude.
By the way, the string on the globe method also works with a non-circular forbidden zone (and with some care it will work on a gnomonic chart, too). For example, if there is some ocean border that can't be crossed for environmental or political reasons, you can still pull a string between starts and end points with "pins" (real or imaginary) marking off zones of arbitrary shape. You'll get the path of least distance with those constraints (in many cases, just as simple as touching a corner, but not always). I can't think of any easy way to extend this to a variable cost function where we assign a weight to a higher latitude, as I described in an earlier post. I can think of some "messy" and mentally entertaining ways of doing this (!), but nothing that would be easy to implement in analog computation. For that general case, code is definitely better.