# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: The flat earth notion**

**From:**Steven Wepster

**Date:**2003 Nov 10, 11:14 +0100

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.