[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/

Thanks Armond for the information.

Am I right in assuming "rhumb-line" is therefore the shortest distance
between two points on the surface of a sphere?
I think the word is being mis-used in this thread and would like an
authority to comment.

However leaving that aside we should be thinking of the most desirable
course to sail and this may or may not be the direct course to the
destination. To give an example, I crewed on an American yacht last century
from New Caledonia to Bundaberg in Australia. After leaving Noumea the
skipper put a line on the chart direct towards Bundaberg. "We follow that
line" he said. Well there is a current that runs south down near the coast
of Australia. The "shortest" distance by pencil did not equate to the
shortest distance in a yacht. We started off heading west and as we got
nearer our destination we were heading WNW. I partly blame his background -
fast power boats - for this situation.

Cheers again

jofra






"Armond Perretta" wrote in message
...
[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/

On Fri, 8 Apr 2005 15:00:30 +1200, "Jofra" wrote:

Am I right in assuming "rhumb-line" is therefore the shortest distance
between two points on the surface of a sphere?
I think the word is being mis-used in this thread and would like an
authority to comment.

No. A great circle is the shortest distance between two points on the
surface of a sphere. But a rhumb line is plenty good enough for the
distances on most Mercator charts.



Rodney Myrvaagnes NYC J36 Gjo/a


"Nuke the gay whales for Jesus" -- anon T-shirt

Jofra wrote:

Am I right in assuming "rhumb-line" is therefore the shortest
distance between two points on the surface of a sphere?
I think the word is being mis-used in this thread and would like an
authority to comment.

I don't know for sure that any authorities are reading or writing to this
thread, so I'll substitute until one comes along. No, you are _not_ "right"
in your assumption. Re-read the definition posted earlier.

However leaving that aside we should be thinking of the most
desirable course to sail and this may or may not be the direct
course to the destination.

If you were to say "the fastest course is the best," or "the most
comfortable course is the best," or ... etc., etc., then perhaps there would
exist some basis for discussion. Otherwise it is very difficult to make
progress on this subject using vague generalities. In addition I am still
under the impression that the discussion concerned steering error rather
than piloting technique.

... To give an example, I crewed on an American yacht last century from
New Caledonia to Bundaberg in
Australia. After leaving Noumea the skipper put a line on the chart
direct towards Bundaberg. "We follow that line" he said. Well there
is a current that runs south down near the coast of Australia. The
"shortest" distance by pencil did not equate to the shortest
distance in a yacht. We started off heading west and as we got
nearer our destination we were heading WNW. I partly blame his
background - fast power boats - for this situation.

I cannot say for sure that I understand your example since there are quite a
few details that have been omitted, but what you _seem_ to be saying is that
no allowance was made for current when setting a course. As mentioned
earlier, this has little _direct_ relation to steering error, although using
the broadest definition of "steering error" would include in many cases the
effects of unidentified or unquantified currents.

I don't know that anything I have written has been much help to you, but I
am afraid that I've pretty much run out of ideas at this point, old sport.

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