This image may help
Green is the starting point, red is the target, the brown square is the obstacle, and the grey areas are the areas blocked by the obstacle.
Note that there is only one target and one obstacle. The target and obstacle are repeated to show what happens when you go off the right edge of the grid, and wrap back to the left edge.
You can see that each time the line wraps around the grid, the angle to the target, and the angles to the obstacle are reduced. Eventually, the obstacle starts shadowing itself. Beyond that point, there's no hope of ever reaching the target. So in this example, there are exactly two angles that reach the target.
Add another obstacle (purple), and all angles are blocked even sooner.
If the obstacle is at the same level as the target, it takes longer before all possible angles are blocked. But eventually the obstacle will shadow itself, and all target angles beyond that point are blocked.
And just for completeness, obstacles above the target can be ignored. The angle to the target will always be smaller than the angle to the obstacle.
Because this is a torus and not a cylinder, objects above the target must also be considered as well as objects below the target if the target is below the start as the bottom edge leads to the top edge. The idea of the object shadowing itself eventually is very helpful though.