NavList:

I think we need to define precisely what the problem is before coming up with any more solutions.  As I see it, the problem initially asked was if you start from A and steer a set course for a set distance where will you end up?  The method you might use to ascertain this really depends upon the latitude you start from, how far you intend to travel, your heading source, and if you’re using a chart, the map projection involved.  For example, close to the equator you hardly need to worry about the difference between a minute of longitude and eastings, because even at 10° latitude, secant10° = 1.015.  Close to the poles, longitude is almost unusable, because secant 85° = 11.474, so you need to employ ‘grid’ navigation.  Therefore, let’s assume we’re operating in temperate latitudes.    

As Frank says, up to about 100 miles distance, it hardly matters which method you use, because such a small portion of the Globe equates very closely to a flat Earth.  Be careful with charts however.  With a Mercator’s chart, the bearing from B to A will be the bearing from A to B + - 180.  With a conical chart, it won’t be, because of the convergence of the meridians.  To measure the bearing from A to B, the practical navigator will measure the bearing at the mid-point of the line joining A to B.  Above 100nm distance, there might be some value in looking to spherical geometry or even elliptical geometry for a solution (spherical geometry makes my head ache; elliptical geometry makes it fall off!).  The result would be based upon great circles and would be a great result in the classroom, but could you actually fly one or sail one? 

To follow a great circle (other than the Equator and the meridians) would require constant changes in true course because of the convergence of the meridians, so a GM compass in ‘True’ or Mag wouldn’t be much good.  A free running gyro monitored by astro might work (I think.  It’s a long time since I tried it, and only a couple of times), but that’s pretty complicated stuff.  Therefore, there’s not a lot of point getting into serious spherical geometry calculations if you’re actually going to follow a rhumb line over the ground.  What we actually relied upon were approximations to great circles consisting of a number of straight sections flown as rhumb lines on Mercator charts using a magnetically monitored gyro compass.  E.g. from UK to Goose Bay, Labrador we flew NW for a bit, west for a bit, then SW for a bit.  It also took us within radar fixing range of Rockall and the Southern Tip of Greenland, but that’s another story. 

So to summarise, roughly what lat & long are these waypoints, how far apart are they, and if they are a long way apart, which heading source are you intending to steer by? DaveP