This is a problem that arose from a consulting contract with the welding industry. The question is this: How does one control a robotic welding torch so that the welder follows precisely the joint between two cylindrical pipes? A typical pipe weld is pictured in the photograph in Fig. 1, where the joint between the two pipes is clearly seen to describe a closed curve. In the most general situation, the company must be able to form welds between pipes with differing cross-sectional radius and arbitrary joint angle.