Required mathematics for computational geometry research career
I'm going to answer a slightly different question (as indicated in my comment): 


  Is there specific mathematical background that would help prepare for
  a graduate degree in TCS that focuses on computational geometry?


I have found (from conversations with other TCSers) that some kinds of geometric intuition are unfamiliar, so the best kind of background will help you in that regard. A comprehensive list along these lines would include:


Topology (point-set, combinatorial and algebraic)
Differential geometry (and some basic Riemannian geometry as well)
Convex analysis (especially duality)
Some basics in algebraic geometry (at least polynomials)
Basic functional analysis (normed spaces, Hilbert spaces, etc)


All of these tools show up in computational geometry nowadays on a regular basis, and their use will only get more sophisticated with time. 

p.s As an aside, if you have a computer engineering degree, it's possible that your background in core theoryCS is weaker than if you had a straight-up CS degree (because of the emphasis on the EE side of things). If that's the case, then the material in the links provided by Kaveh will be immensely useful, probably more so than this list.