Wow, tsuwm--cool!
The early navigators soon gave up great circle navigation ("orthodromie" en français) as a bad joke when they realised that, when sailing along agreat circle, the bearing changes constantly. They soon decided that sailing along a constant bearing, whilst perhaps taking a little longer, was far simpler. You just measured the bearing on the Mercator's chart, and this was the bearing to follow to go where you wanted to go.
This line of constant bearing is called a Rhumb line. The word "rhumb" (or sometimes rumb and it is the same in French though not very well known) comes from the name of angle measurement representing the "point" on the old fashioned compass cards. There are 32 "rhumbs" in 360 degrees, hence a rhumb is 11 1/4 degrees. From:
http://members.ozemail.com.au/~jjjacq/sundry/navrhumb.html

I didn't look long enough to find a good explanation of loxodromic curves, but take a gander at this title in a list of articles I found: On unsmoothable diffeomorphisms. Bulletin of the American Math Society, vol. 81, p. 746, 1975.

tsuwm, do you have an interest in maps, or is this just yet another evidencing of your vast storehouse of knowledge?