Can you make a graphic with the sun 93 million miles away and straight light path? Can you make a graphic of a globe map where all the distances and direction match observed? There are many many of these, I think you can. RE geometry works with straight light rays to explain day/night. Works without unexplained light bending.
The circumference of 80 degrees north or south is much less than at the equator. How does this project/transform/??? to a cylinder? The circumference at every latitude would be the same as the equator. To a disk? That's why Australia is always too large on the FAQ maps. This is possible on a sphere, how can 80 degrees north circumference be so much smaller than 80 degrees south on a disk FE? They should be equal. Per Gauss, you can't project, transform or
onto a cylinder or a disk and have both 80 degree circumferences be correct.
Your graphic of disk earth has Australia too big, just like the FAQ maps. Let's see your flat map with accurate size, distance, direction, and constant scale. You have math and graphics skills. Let's see it. Be the first!
Re day/night bendy light, do your equations account for the left/right bending? At noon on the equinox in Kampala, Uganda, the sun is rising in Rio De Janero. It appears to be directly east bearing 90 degrees, but on FE disk map, it looks like the bearing is actually about 45 degrees. And from Perth Australia at the same moment, looks like 270 degrees (directly west), but is actually 315 degrees.
Also interesting is that the light has no left right bending directly north and south. The amount of bend increases as you go south along the line of sunset until it gets to the far corner of the lighted area, where the discrepancy is maximum, then decreases until it is again 0 directly south.
So you need to add that to your model.
Interesting physical phenomenon, bends the opposite way on either side. As the sun moves around, this pattern moves with it. Apparently, the sun throws curveballs?
Now add the dome to your model. When it is sunset in Denver, in Salt Lake City it is still light blue over the entire dome. In St Louis, it is dark and there are stars over the entire sky, including the part of the dome beyond Salt Lake City. If someone in St Louis and someone in Salt Lake City look at the exact same spot over Denver, one sees light blue day sky, the other sees dark night with stars.
So let's see a model that accounts for left/right bending, and shows how someone can see either the entire dome of light blue at the same time as another person a few hundred miles east sees dark and stars, and has circumference of 80 degrees latitude both north and south much smaller than the equator on a flat disk, or cylinder, or anything other than a sphere.
You claim to have the ability to produce a FE map with accurate distance, size, direction, and constant scale. I await your post and will be astounded.