A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David Pike
Date: 2021 Dec 28, 15:28 -0800
Tony Oz you wrote: "I'm suprised to read that there are difficulties with plotting a track aross the equator on the Mercator projection."
There isn’t a difficulty. It’s proving the thesis that, if the great circle is north of the rhumb line in the Northern Hemisphere and south of the rhumb line in the Southern Hemisphere, then the great circle and the rhumb line must cross the Equator at the same longitude that concentrates the mind.
If indeed true, it’s hard to show perfectly using a numerical example, because meridional parts tables relate to the terrestrial spheroid and spherical geometry relates to a perfect sphere. Moreover, the crossover longitude in the great circle case relies upon using information calculated in earlier equations, so rounding errors start to multiply, particularly at angles where trig functions change rapidly.
A closer match of GC crossover longitude might be gained using ellipsoidal geometry and the terrestrial spheroid. Another method might be to consider the intersection of three discs, one containing the great circle through the start and finish points, one containing the Equator, and one containing the Greenwich Meridian. I find these methods hard to imagine but I think there’s some scope in the discs idea, especially if I can show that the GC disc crosses the EQ disc in the ratio of the longitudes of the start and the finish points. Then it’s just like a 3D Mercator Sailing. DaveP