1. Intersecting cylinders problem.....

OK....ran into an interesting problem at work that is pissing me off.......

1) Imagine a vertical cylinder 6' in diameter and say 10' in height

2) Run two horizontal cylinders (say 12" diameter and 36" diameter) through the larger cylinder but at different elevations....I'll say the 36" cylinder axis intersects the larger cylinder axis at a point 4' above the bottom of the larger cylinder and the smaller 12" cylinder axis intersects it a distance of 8' above the bottom......

3) I'll also say that the angle between the horizontal cylinders is 90 degrees as viewed from the top.....remember though that the elevations are different....

The question: What is the minimum distance between the horizontal cylinder walls as measured along the larger cylinder wall?

More clarification: The situation described is a manhole with pipes penetrating the manhole walls at different elevations and with different angles (I just picked 90 degrees for simplicity). For a designer, the distance in question can only be so small before the integrity of the concrete manhole wall is compromised. While there are empirical guidelines to assist designers published by various agencies, I was wondering if this has a mathematical solution. I tried but suspect this may involve some calculus that I have forgotten. The curved path is not circular so that is what is throwing me off.

Anybody?

2. Here, I made a sketch showing what I'm after:

Cylinders.pdf