UCLA Math/Computer Science 1975. Never used the math because I became a software engineer. I can barely remember the math, but I learned stuff like what a coordinate system is and can refresh my math memory with internet. I remember enough to see the wrong math you do when make it complicated in order to hide the wrong part. Arguing details is pointless, you are not here to learn, you are here to justify FE.

I talked about FE with a programmer friend. He said that he had written the nav software for the Canadian Air Traffic Control system and used 3d geometry, haversines, straight out of a math textbook. Worked perfectly, airplanes in Canada today arrive exactly where they are supposed to, FE math would be different and not work. He has a degree in math.

I would be astounded if there was a qualified mathematician or physicist that agreed with any of Troolon's math. I am not surprised that he cites secret experts. Confirmation without possibility of falsification.

I remember when my linear algebra prof started the lecture with "Today we are going to talk about how we know the earth is not flat, Gauss' Remarkable Theorem." The normal vectors on the surface of a flat disk are parallel, so curvature is zero. The normal vectors on the surface of of a sphere are not parallel, so the curvature is not zero. Find me a math prof who will disagree with this. There aren't any. There is a reason why you are sayoing this on FE site, say it on a math web site, or astronomy, or astrophysics.

What you did is simple, you peeled the surface of the sphere off from the south pole and stretched it out into a disk, like popping a balloon and stretching the balloon out into a disk. All the rest is just blather. Doing this stretches out the size of everything in the southern hemisphere. In your graphics, Australia is clearly bigger than North America. Measurement is measurement, and there are many ways to know the true sizes.

But distance is not your only problem. When you stretch it, the south pole becomes the circle around the edge. If you incorporate Sigma Octantus, as directly above south pole, it becomes a circle rather than a particular point. Even if this made sense, you have to explain why observers in the southern hemisphere see it as a small dot on the part of the circle and do not see the rest of the circle. A difficult to explain combination of bent light and directional light, much like the spotlight sun problem, but worse.

Except for one thing, Sigma Octantus is not directly above the south pole, it is a little over a degree off axis. So consider the Southern Cross. It is enough south that it is seen from everywhere in the southern hemisphere as being due south. It is much like the big and little dippers in northern hemisphere. Where is the Southern Cross? It appears everywhere as outward from the disk. Where is it really? That question has no sensible answer on FE.

According to your theory, since everything is equivalent, seems like I should be able to do the same math conversion using the south pole. This gives Sigma Octantus pretty close to observed re azimuth. still requires bent light ("unknown forces with unknown equations", per the FAQ) for altitude. But now Polaris, the north star, is everywhere around the edge. Also seems pretty arbitrary to start with the poles, how about your house in the center? Start with your house and peel the surface of the sphere starting with the point directly opposite on the spherical globe. The same transform can be done choosing any point at the center. If you choose a pole, one of the pole stars makes at least some sense. Start at the equator and neither makes any sense at all.

You made graphics to show a transform to several different shapes. Include the pole stars in these transforms and see how much sense it makes. Let's see graphics that include the Southern Cross and the Little Dipper, visible from the places and in the direction that matches observations. You can't do it. Distances, direction, and observed location of astronomical objects are all correct in RE graphics. They are not correct in any other shape.