Are your parts represented in two or three dimensions? If they're 3d meshes, you can use the same techniques as others have described, but you'll need to do polygon/line intersection instead of line/line intersection. Either way, if you're doing a lot of tests like this, you might want to build a BSP tree out of your data. BSP tree intersection tests are average-case logarithmic time, and by building the structure once you might save yourself some computation.

--
F o x t r o t U n i f o r m
Found a typo in this node? /msg me
The hell with paco, vote for Erudil!