I haven’t forgot my little problem about working out where the apparent edge of the world was, from my visit to the Sleeping Bear Dunes in northern (lower) Michigan. What I have been is stuck on a way to do all the calculations in a way that’s clear and that avoids confusion. I realized the calculations were reasonably clear to me but were hard to describe because I could put into similar-looking symbols a bunch of things I wanted to describe.
So I’ve resolved that the best thing I can do is take some time to describe the things I mean, and why they’ll get the symbols that they do. The first part of this is drawing a slightly more mathematical representation of the situation of standing on top of the dune and looking out at the water, and seeing the apparent edge of the dune as something very much closer than the water is. This is what’s behind my new picture, a cross-section of the dune and a person looking out at its edge.