This chart shows the great circle arcs across a sphere but mapped onto a pizza pie chart. Notice that the forward direction follows the curve of the circle. Follow the curve with your finger and the direction of the curve should also be the same direction. What's strange is how the curve bends and warps but the video game should render that bendy curve as the same direction. We need to figure out how the direction of the curve changes relative to the x y flat plane the game is projected onto. For example: notice how the middle of the chart is a straight line with very little change in direction, that will result in the north pole having a small change in direction. The edge has a large change of direction, that will result in the south pole having a large change in direction. I think the game renders sprites in x y flat plane which contains the projection of a sphere, this is the source of the error. The AI and physics works correctly so the root cause of the rotation bug is the projection. Projecting position works fine, Projecting rotation does not. This is the most difficult part to make this game work, once this is solved, it's a matter of adding content.

If we add the rotation/angular/orientation gradient/derivative to the angle, we might approximate the rotation bug, this would require storing two positions. The two positions are projected onto this pizza pie chart and the angle between the two points are computed. This correction value is then added to the angle.

Added the Theta Correction Chart to experiment with multiple polar charts to make the transition between hemispheres more smooth. The original code had an error where when you moved near the south pole with a polar distance less than 0.1, it would throw values as large as 90 degrees. That would force the spaceship to rotate 90 degrees side ways when approaching the south pole. The goal is to avoid hard changes which are shown in red highlight. When you're near the poles, the table will bold the text.