[re-posted]

Jofra wrote:

To advance this discussion would you mind setting out your view as
to what a "rhumb line" is.

Sure thing, but you should realize that the information below isn't merely
or necessarily "my view."

A rhumb line (or loxodrome) is a path of constant bearing on a spherical (or
elliptical) object. The word "loxodrome" comes from Greek loxos : oblique +
dromos : running (from dramein : to run). They are a traditional part of
the theory of navigation.

If you follow a given (magnetic-deviation compensated) compass-bearing on
Earth, you will be following a rhumb line, which spirals from one pole to
the other. Near the poles, they are close to being logarithmic spirals (on a
stereographic projection they are exactly that), so they wind round
each pole an infinite number of times but reach the pole in a finite
distance. The pole-to-pole length of a rhumb line is (assuming a perfect
sphere) the length of the meridian divided by the cosine of the bearing away
from true north. Rhumb lines are not defined at the poles: it is hard to go
south-east from the North Pole and even harder to go north-west.

On a Mercator projection map, a loxodrome is a straight line; beyond the
right edge of the map it continues on the left with the same slope. The full
loxodrome on the full infinitely high map would consist of infinitely many
line segments between these two edges. On a stereographic projection map, a
loxodrome is an equiangular spiral whose center is the North (or South)
pole.

Finding the loxodromes between two given points can be done graphically on a
Mercator map, or by solving a nonlinear system of two equations (details
omitted for brevity). There are infinitely many solutions; the shortest one
is that which covers the actual longitude difference, i.e. does not make
extra revolutions, and does not go "the wrong way around."

The distance between two points, measured along a loxodrome, is simply the
absolute value of the secant of the bearing times the north-south distance
(except for circles of latitude).

Old maps do not have grids composed of lines of latitude and longitude but
instead have rhumb lines which a directly towards the North, at a right
angle from the North, or at some angle from the North which is some simple
rational fraction of a right angle. These rhumb lines would be drawn so that
they would converge at certain points of the map: lines going in every
direction would converge at each of these points.

Hope this is what you were looking for.

--
Good luck and good sailing.
s/v Kerry Deare of Barnegat
http://kerrydeare.home.comcast.net/