look at what you're (probably) dealing with in this situation first (speaking from personal experience):
your shyness
lack of confidence
lack of experience
the fear of initiating a new experience on your part.
the fear of being rejected
the fear of the unknown
the fear of being publicly humiliated in front of the class.
not knowing what her reaction will be no matter what you say or how you say it
the fear of being dumbfounded (brain-lock) in a highly emotional and sensitive encounter
the thought that maybe not knowing if she likes you is better than finding out that she actually hates everything about you
the fear that if you do nothing, you will regret it for the rest of your life
etc.etc.etc.etc.etc.
all of those fears can be overcome by:
your self-confidence
your optimism
your sincere expressed desire to get to know this beautiful girl
ask yourself:
what do you really want from her?
what realistic results can you expect from getting to know her?
is imposing yourself on her feelings of personal privacy worth the risk?
does her body language tell you that she is generally friendly and sociable?
if after this short introspection you still feel that the rewards outweigh the risks, which from what i gather you definetly do feel, then before or after class just walk up to her and introduce yourself and tell her that you find her really interesting and would she mind if the two of you could become friends? (as a safe harmless way of gradually entering into a relationship, rather than the "in-your-face-take-me-or-leave-me-approach")
you'll never know unless you give it a shot, so what're you wait'in for? go for it.  

*just be emotionally prepared for any and all eventualities- because you can't make up her mind for her*