Why do I need to paint/vacuum the corners when creating a mowing zone?

The painting/vacuuming technique is used for mapping corners of your lawn where it is impossible to completely map the corner in one motion.

Your mower uses the edge of the mower while mapping, and it may take multiple passes to completely map the boundary. By pushing the mower through the corner at different angles you can create a boundary that is as close to the edge of your lawn as possible.