How can I approach a girl whom I have never met or with whom I have never spoken but with whom I have experienced eye-to-eye contact whenever we see each other?
"Eye-to-eye" contact can provide an opportunity to approach a girl whom you find physically attractive and, hopefully, intellectually-engaging. That eye-to-eye contact, however, would, optimally, also involve some form of facial expression on the part of the girl that suggests she is open to a more formal meeting. For example, does she smile or, at a minimum, present the appearance of being interested in getting to know you? The slightest suggestion of a grin on her part could indicate that she is as interested in getting to know you as you are in getting to know her. Absent any kind of overt facial expression, the repeated eye-to-eye exchanges can still definitely provide an opportunity to approach this girl and introduce yourself. There is absolutely nothing wrong with walking up to her and introducing yourself and seeing if she is interested in engaging you in conversation. If you are aware of any shared interests, such as sports, music, or whatever, that can help 'break the ice' and facilitate a first conversation.
While it is possible to gauge the girl's interest in you through mutual friends, that is not the best way to go about this. She will respect you more if you demonstrate the courage and self-confidence to approach her and introduce yourself. If she is receptive to your overtures, such as by immediately and happily engaging you in conversation, then you are on your way to developing a relationship. Be advised, however, that, if she reacts in an unfriendly way, and that is possible, you should be prepared to drop the matter and move on to other pursuits. You don't want to try and force the issue, but you do need to be assertive in simply introducing yourself and maybe asking if she is interested in meeting you at a nearby hang-out, or something along those lines.