Most animals, including big cats, seem to regard direct eye contact as a threatening gesture. How they respond to it depends on the situation, the size of the animal, and the size of the "threat." A mountain lion is very unlikely to attack an adult human, even in self defense, if there is an escape option, so eye contact (combined with other threatening gestures) can be an effective way of scaring it off. This approach easily could backfire if attempted by a child. A bear is more likely to respond to eye contact by charging. If you wish simply to observe animals in the wild, you should try to avoid direct eye contact.