Funny you should mention critters, as last night I snuck up on both a ground-hog-ish critter, and later a rabbit in the backyard. My wife's frustration with their nipping away at her gardening had me in slow/sneaky pursuit with a broom in hand(s).
I definitely could have "cessated" the former, I think mostly due to its eyes being more in the front of its head than on the sides. I definitely got closer to it than the length of the broom before it noticed me and froze.
But I just didn't have the heart to go through with it in that moment, opting for an "instilling of the fear of God" substitute instead.
The rabbit, well, those eyes on the sides of its head meant having to slowly move and pick up feet, generally trying to conceal the lifting of a foot behind the calf of the other leg. It caught on pretty quickly, but seemed to have quite a bit of patience in its seeming "maybe it doesn't see me" tactic.
I followed its trotting away several times until it finally did so an order or two of magnitude more quickly, finding an opening under the fence before I could see it disappear through it.