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In reply to Jan Kalidova's question:
>Could anybody give the analytic equation for the length of the loxodrome
depending on the initial latitude, the latitude difference and the course?
The distance along a loxodrome following a course K between two parallels
of latitude, is the distance between those parallels divided by cos K (if
the cosine is negative, neglect the sign). You may check that the relation
holds for K=0 and K=180, while for K=90 or K=270 the answer would be
infinite, indicating that you will never reach the other parallel.
Surprisingly length is independent of the latitude of departure; only the
latitude difference and the course matter.
In particular, for any course other than due east or west, the loxodromic
distance from any latitude to any pole is finite.
I stumbled upon this strikingly simple formula when I evaluated the path
integral of the loxodrome in spherical coordinates. That might sound
complicated, but it isn't. I assumed the earth is a perfect sphere.
George pointed out that at the pole, our vessel would be spinning at
infinite rate of turn. That sounds worse than moving at infinite speed. It
means that we not only need an infinite amount of energy, but we would get
infinitely dizzy, too. Perhaps there is a way to increase the rate of turn
at the cost of forward speed, but I presume that we would in the end
approach the pole infinitely slow. My physics is not sound enough to
understand this completely.
Steven.
