when walking, why do cats walk diagonally across one's path both on a staircase and on level ground?
I don't believe anyone has done a scientific study on this matter, but as the owner of three cats, I certainly do recognize the phenomenon you are asking about!
Horses, deer, and other grazing animals typically have eyes located on opposite sides of their skulls, an arrangement which means that, while they have virtually no depth perception, they do have a very wide field of view and can see almost all the way around themselves without having to move their heads or roll their eyes to do so.
By contrast, cats and other predator animals have their eyes located in the front of their skull like humans do. This allows for good depth perception, which is essential for pouncing upon or manipulating things. The trade off for good depth perception is a narrower field of vision. Humans have a field of vision of about 180 degrees. Cats, because their eyes sit farther forward in the sockets, can see a bit further to the sides, so their field of vision is around 200 degrees. This means that there is a 160 degree field behind them that they cannot see without moving their heads. Cats, like most hunters, tend to be very attuned to what is going on around them. I believe they walk diagonally so they can more easily keep track of what's happening behind them. Most house cats have learned the hard way to watch out for human feet.