The rules of eye contact are simple. After catching her eye:
If she looks down: She’s instantly attracted but shy.
Approach this girl, but go easy on the cocky jerkboy game. She’s a natural introvert, and a romantic at heart. Don’t come on too strong. A light touch will do, flirty and coy. She’ll just be happy you even had the balls to escort her from her dreamy inner sanctum.
If she looks to the side: She’s not instantly attracted.
You will have your work cut out for you on this girl. A side-looker is as good as (or bad as, depending on your perspective) a cold approach on a girl who hasn’t noticed you. The side-looker has a boyfriend, or she doesn’t like your look or your leer, or she’s a manjawed feminist who is constitutionally incapable of flirting with men without having an existential moral crisis. You can turn a girl like this around, but it will mean you have to be exceptionally bold and full of teasing and negs. An effective opener would be one that immediately disqualifies her, flips the script, and assumes the sale. For example, “I caught you checking me out. Don’t worry, even though it’s nothing new, I’m still flattered.”
If she holds eye contact intensely: She’s instantly attracted and slutty.
Weaker men wilt under the pressure of the hard-eye contact girl. She’s dripping sex from her limpid orbs, and only men made of sterner stuff will rise to her fightin’ iris challenge. Nothing much needed here but an open-faced “Hi” and a pretext to absolve her nascent feelings of aggressive sluttitude (such as asking her for directions if you cross her path on a street corner). A direct, “Hey, I noticed you from across the room, and had to come over and see what your deal is”, will work in any bar setting.