"How does illuminate the world?" What are you talking about? Please use proper grammar.
You should know what I am talking about, even with the typo, as I have raised the issue before.
Again, hypothetically when the sun is at its southern most point, taking south as the bottom of your map and north as the top, how does the northern region get illuminated while the region in the middle is in darkness?
It makes no sense at all.
I have brought up this issue before and you just repeatedly deflected then ran away.
Here is an image, derived from the one provided on your site:
Please explain how Alaska/West America, Eastern Russia and Australia/NZ have daytime/are illuminated by the sun, but Africa and England are not.
Please draw in your nice roughly circular patch of light.
I can draw one in quite easily thanks to your longitude lines:
Notice how instead of a circle of light we get a circle of darkness?
But don't worry, you can still get the semicircle of light/dark and various other shapes.
Under the Bi-Polar model the sun's area of light is more circular in the Bi-Polar model all throughout the year.
No it isn't. You just push the problem around. You still have the ridiculous patches of light.
All it does is change the timing of when the ridiculous patches of light are and for how long. It doesn't solve the issue. But yes, it does manage to get a nice circular patch of light for one point of the year as it is an azimuthal equidistant projection of the real round Earth, centred on a point that sun can be directly above.
The path of the sun does retrograde throughout the year.
If you mean it changes it's apparent position, yes, it does. But that doesn't mean it takes a figure of 8 path.
Notice how they can't explain it with regular straight line geometry between the sun and earth and go with "The spreading of the trails as they go upward is a distortion caused by stretching the domed sky onto a flat semicircle."
No, they can explain it. They are just stating that projecting an intrinsically non-flat image onto a flat surface results in distortion.
We have also been over this.
Projecting an angular based view onto a 2D image will result in distortion.