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George,

I know better than to question your knowledge of the mathematics of
navigation. I also strongly suspect that, if I asked for proof that a
loxodrome reaches (rather than approaches) a pole, you would need to
resort to mathematics that would not easily be represented in e-mail
format and which I would not understand if you did send it. Hence, I
have little option but to accept what you say at face value -- something
that I am never comfortable doing, even when you are the author of
whatever I am accepting.

Could you, perhaps, explain the final approach of a loxodrome to the
pole, without resorting to math? Above a latitude of, say, 89 degrees 50
minutes, the sphericity of the Earth ought to be negligible and we
should be able to visualize the problem as one of a rhumb line
spiralling around the pole in a simple 2-dimensional space, while
cutting each meridian at the same angle. I can visualize that spiral
closing in from 10 miles out to a mile and so to a tenth of a mile. But
it then seems to me that we could simply expand the scale by 100 times
and see the loxodrome spiralling in from 1/10 of a mile to 1/1000,
before expanding the scale again and repeating. What I don't see is how
that loxodrome will eventually make the final step of reducing its
distance from the pole to zero.

You wrote:

> They are both wrong, and Herbert is correct; though I'm sure he is capable
> of defending his corner without my aid.
>
> The presumption is that you are steering a rhumb-line course, with a
> Northerly component, and that all the Polar ice has melted.
>
> As you spiral in toward the pole on a rhumb-line course, you travel a
> FINITE distance to get there. If you can maintain a constant speed, then
> that can be done in a finite time. The snag is, you have to travel in
> ever-decreasing circles, as the size of the spiral shrinks. To get exactly
> to the pole, you have to make an infinite number of such gyrations, so the
> vessel has to be spinning at an infinite rate. This is rather an unphysical
> state of affairs, to say the least: a "singularity".


Maybe it is my limited understanding of math but I don't think that the
finite length of a loxodrome is inconsistent with it never reaching the
pole. The mathematics of infinity has some funny properties, analogous
to the effects of dividing by zero, and I'd not rule out a curve of
finite length which spirals without end. (There may well be members of
this list who can say that I am wrong. I'm only saying that I can't rule
that out myself.)

> The pole will be reached at a predictable moment, at which the ship will be
> spinning round at infinite speed. Just after that moment, it will escape
> from the pole, still spinning with infinite speed, and the radius of the
> spiral then increases, and the spinning slows, until the vessel reaches its
> original latitude. What will its longitude be then?


While I am willing to accept that I may be wrong and a loxodrome may
actually reach the pole, it will take a whole lot more to persuade me
that said loxodrome ever departs from the pole again. George: I really
think you are wrong on that point.

I _know_ you are wrong on the claim that our hypothetical ship will ever
leave the pole. You have assumed that the vessel is following a
rhumb-line course between 270 and 090. Any course away from the North
Pole must be 180 instantaneously and, immediately thereafter, must be
between 090 and 270. That is: the hypothetical ship can only leave the
pole by changing its course (perhaps to its reciprocal) and that is
contrary to the starting assumption.

> Of all the unrealistic questions we have considered on this list, this is
> perhaps the most unrealistic of all. But why should we let that deter us
> from playing such games?
>
> The picture above, of a vessel spinning infinitely fast at the pole,
> applies to all incoming courses except 0deg and 90 deg, as Trevor points
> out. At 90deg, the pole is never reached at all: the vessel sticks to the
> equator.


No. That is commonly stated in textbooks but is clearly false.
Regardless of its starting latitude, a vessel following a rhumb-line of
090 or 270 will never reach either pole. It will follow a parallel of
latitude, which could be the Equator but could be any other. (This is as
stated in my last contribution to this thread.)

> At 0deg, the vessel approaches the pole along a certain line of
> longitude, then emerges along a line of longitude 180deg different.


Again, the vessel cannot leave the North Pole while its course remains
the rhumb line of 000. As I wrote last time, it can only leave by
changing to a reciprocal course -- which it would of course do if it
continued "straight" (meaning on a Great Circle, since this is spherical
geometry) through the pole. We are, however, assuming rhumb-line
courses, not reversible rhumb-line courses.


Trevor Kenchington


--
Trevor J. Kenchington PhD                         Gadus@iStar.ca
Gadus Associates,                                 Office(902) 889-9250
R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555

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