This paper reports on research efforts to efficiently find a Voronoi tree for cavities described by Non-Uniform Non-Rational B-Spline Curves. It examines the motivation for solving this problem, and it reviews the literature on splines and their hodographs (or offset curves). This paper then offers a technique for finding a Voronoi tree for cavities by intersecting hodograph curves. The possibilities for an analytical determination of intersection are considered, and a numerical methodology is presented. This numerical method is demonstrated on two different types of problems: a symmetric cavity problem and an asymmetric cavity problem. Finally, conclusions are drawn about this work.