Okay, making progress...
Then I almost freaked out at finding the midpoint. Seriously, not a proud day in the Kaolin life. But I got that with just a short bit of panic and a shake of the head. Now pounding my skull on the nearly-as-obvious disk/sphere collision. I have an intuition for how to do that that feels fairly inspired, and I just need to figure out exactly which variables to tweak to get what I need.
