What is the relation between Computer Graphics, Discrete Geometry, and Complexity Theory? I am a master computer science student, and I am interested in both geometry and complexity theory.
So I would like to know what is the relations between discrete geometry, computer graphics, and complexity theory ? 

I know the answer partially but I would like to have a clear answer since I do like to work on these fields in the future.

Thank you in advance.
Have a look at this slide of David Eppstein in which he tries to draw connections between these subfields. In my opinion, a computational-geometric complexity result is likely to be confined to computational geometry itself without broadly affecting complexity theory in general.

Excerpt from the above slide: Some computational geometry problems have practical applications. Their theoretical hardness are all conjectured to be $NP$-hard but still have not been proven. Some have approximation schemes.