An Application of Analytic Geometry to Designing Machine Parts--and Dresses

This paper presents the solution of an engineering problem that the author was asked to solve. The problem involves creating a flat pattern that could be cut from a piece of sheet metal and rolled to form a tube whose top edge would be contained in a plane that is not perpendicular to the central axis of the tube. A piece of this nature needs to be fabricated whenever two sheet metal tubes must be joined at any angle other than a straight angle. My brother-in-law called me with an intriguing yet surprisingly simple problem one evening about 20 minutes past 11:00. It was not the first time he had called me looking for help with a math problem, nor the first time that he had called me so late. Still, I was tired, and more than a little annoyed with him. Despite that, being asked for help with a problem has always managed to touch a little corner of my mind that is quite proud of being a mathematician. So I listened. James is a mechanical engineer who works for a firm that designs commercial bakeries. In the course of redesigning a machine that was not working properly, he had found it necessary to have a tubular part which would be fabricated by rolling a piece of sheet metal into a cylinder. What made the situation challenging was that one end of the tube needed to be cut at an angle that was not perpendicular to the central axis of the tube. It’s much easier to cut a piece of sheet metal while it is still flat, so James wondered how he would need to cut the top so that it would have the form he desired after the sheet was rolled into a tube. James is accustomed to relying on AutoCAD to find the necessary geometric properties of a design. In this case, he was at a loss as to how to apply this software. So were the engineers that he works with. Fortunately, years of mathematical training as an engineering student had left him with the ability to recognize this as a problem in analytic geometry. He just wasn’t quite sure how to proceed. After James had explained the problem to me, he asked whether the appropriate curve might be found by describing the tube with an equation and finding its intersection with a plane having the correct slope. I had had the same initial thought almost immediately after he explained the problem. Just as quickly, I had encountered the same problem that was perplexing him. While it is easy enough to describe the intersection of a cylinder and a plane with parametric equations, neither of us could see what would happen to those equations if the intersecting curve were “unrolled” onto a plane. After realizing that I didn’t know how to “unroll” the curve, I decided to give some thought to what I was really trying to find. All I needed was a function describing the curve as it would be when the sheet was flat. It occurred to me that in this state, it might very well be possible to describe the curve with a two-variable formula, where the variables represent rectangular coordinates. For reasons that should seem clearer later on, I will call the variables s and z rather than the more standard x and y. The variable s measures horizontally along the http://academic.udayton.edu/EPUMD ISSN: 154-2286 Electronic Proceedings of Undergraduate Mathematics Day, Vol. 3 (2008), No. 5