The Patent and the Formula

A while back, Uncle Skip gave me a bloggity award, the acceptance of which entailed me telling you all a few things about myself. And so I did, without actually telling you much at all about myself. Except for one thing - I mentioned that I have a patent to my name, and possibly a mathematical formula. Which struck a few of my commenters (I'm lookin' at you, Bijoux, Sailor and Lime) as interesting, and they asked me to give more details. So I left my own responsive comment, giving the short version of the two stories. And, in the ensuing months, I've come to think that those stories might be worth a full-blown post, giving a fuller account. You know, for posterity, and all that. (See, I'm just vain enough to think that somebody, somewhere, centuries in the future might stumble across this blog, and say, "Wow, this dude had a patent and a formula! I wonder who the hell he was?")

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The Patent

When I was still working at my first job, I did some work on a project that was a bit unusual, for a customer that usually worked closely with one of our competitors. They had an idea for a non-pneumatic spare tire/wheel, to save space and cost (as against the typical pneumatic mini-spare, or even moreso, an actual full-size spare). They had gone first to our competitor, with whom they had a long-standing working relationship, and asked them to develop the design for them. But the competitor was unable to come up with an actual working design, so they came to us.

When we got the design, we could see immediately that, as given to us, it wouldn't work. But our company had had, for many years, contracts with the US Defense Department, to make wheels for tanks and other tracked vehicles, which were also non-pneumatic, and there were a couple of design 'tricks' that made such a wheel actually work.

So when I was given the project, I did my usual analysis and immediately saw that the stresses were way too high. But, I told my bosses, this was really a pretty similar thing, conceptually, to a tank wheel, and what if we applied tank-wheel design principles to it? So I did a re-design according to tank-wheel design principles, and when I analyzed it, lo and behold, we got a working design! So we made a few prototypes and tested them, and they seemed to work just fine. And I, having done my job, was happy, and moved on to my next project.

At that point, our top management got a little nervous (and not without reason), that our customer could just take all our great design/development work and hand it back to our competitor, with whom they had a nice comfortable business relationship. So they decided (unbeknownst to me) to seek a patent, in order to protect our 'intellectual property' rights.

Thus it was that, a year or so after I'd worked on the project, and had all but forgotten about it, I was called into a series of meetings to describe to a bunch of lawyers what I'd done, and blah, blah, blah. And the upshot was, that when the patent was granted, my name (among others) was on it. It had been an interesting project, and I was a little proud of having made the 'conceptual leap' to incorporate tank-wheel design principles, but as far as I was concerned, I was just doing my job. Getting the patent was never my idea (although I was certainly not opposed to it); someone else sought the patent, and in the end, because I'd worked on it, my name was one of the ones that got slapped on it.  So they paid me a dollar for rights to the patent, and I was invited to the annual Patent Luncheon for as long as I worked for the company (which turned out to be just a couple more years).

So now, I've got a patent to my name, which is nice. (Honestly, though - my dad has something on the order of 20 patents, in half-a-dozen countries, so I don't make a bigger deal of it than it merits. . .)

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The Formula

(OK this is math; if your eyes start to glaze over, feel free to stop at any time)

This one comes from my first job, as well. In the course of doing Quality Assurance on our parts, we would scan them in a machine that essentially took a series of points (typically a number in the hundreds, or at least the tens) from the surface of the part. Then we had to take those hundreds of points and generate the key dimensions we were trying to measure from them - a radius, an angle, or what-have-you, in order to determine whether they were within the specified limits, or not. Getting an angle from the point data was pretty straightforward - there are lots of well-understood methods and formulas for computing a line (and thus, an angle) from a set of points (I should note that the key thing here is that we had so many points - if we only had two points, then it's straightforward to calculate the line between them; any middle-school student could do it. But when you have more than two points, to say nothing of hundreds, it gets more complicated to compute the line that 'best fits' the large set of points.)

For a circle, though, there weren't any readily-accessible methods or formulas available, as far as I could tell. From high-school Algebra, I knew that I could calculate a circle from three points, but I had no idea how to do it for five, or fifty, or five hundred points. I played with calculating a circle from every possible set of three points taken from the larger point set, and that gave me a 'bracketed' answer - the limits between which the 'real answer' had to lie - but that wasn't a very satisfying answer. I even went to the Math Library at the University, to see if there were any obscure formulas published anywhere, but they couldn't easily point me to one, either.

So, over the course of six months or so, I played around with the problem, trying to figure out a way to attack the problem. From time to time, I'd have an idea that seemed like it might be fruitful, and I'd make some progress on it, but it was slow. And I still had to work on my regular job, so I didn't just have unlimited time to work on the 'best-fit-circle' problem.

Then one day, I had a flash of insight (it involved choosing a workable error-measure, if any of you are math-y enough to know what that means), and cranked the formula through (it involved a LOT of cranking). Then I wrote it into a computer program, and ran a few point sets through it, and what do you know? It worked! It calculated a circle that by-golly corresponded to the set of points. So, I put the computer program into a more-usable form, and we went on our merry way, using the formula in the day-to-day work of checking radius dimensions on the parts we manufactured.

Now, I didn't for a minute suppose that I had actually derived an original formula. I thought all along that someone, somewhere, had derived this formula years before, and I just wasn't clever enough to track it down. No big deal - many times, in Math classes I'd had over the years, we'd derived the classical formulas that were in our textbooks, just for the instructional purpose of seeing where they came from. And I was sure that this was one of those - I could have looked it up, but failing that, I just derived it myself.

Fast forward 15 years - A guy I know runs a 'Math' column in a little engineering journal, and I'd written a few articles for his column over the years. He was running low on columns 'in the can', and he asked me if I could write another article for him. I showed him my notes for the 'best-fit-circle', and he thought that would make a great article, so I wrote it up, and a year or so later, he published it.

Now, whenever I've written one of those 'Math columns', I always get a few emails in the aftermath of it being published - it's a lot of fun to hash over the column with other interested folks. Once, I wrote an article on how the sun's position in the sky varies over the course of a year, and I got all sorts of interesting emails, from guys wanting to talk about how to design their house to get more sun in winter, and less in summer, or a professor sending me a little 'solar calculator' he'd invented.

When the 'Best-Fit Circle' article ran, I got emails from a couple of pretty high-powered academic-type guys, thanking me for the article, saying that it was obviously really useful, and asking me where I'd gotten the formula from. When I said, sheepishly, that I hadn't found it anywhere, that I'd worked it out myself, and by the way, could they tell me where it might be published, they wrote back, congratulating me for deriving a really useful formula. One guy was on the development team for one of the major commercial computer-graphics codes, and he said he'd searched all through the literature for all the various different methods for computing a circle, and he'd never seen it before. Which kinda stunned me, actually.

Then one day, one of my kids did one of those things that kids will do - he googled his dad's name - and found, among various letters to editors I'd written, and whatnot, at least two instances of college professor-types assigning homework to their classes, involving the formula I'd published in my friend's little journal (properly attributed, and everything). Then, when 4M had a calculus class at the junior college, he mentioned to his instructor that his dad had written a math article, and the instructor asked if I'd send her a copy. She really liked it, and started assigning it to her own classes. I spoke to her once, and she mentioned it to me, and I told her that I couldn't really believe that it was original with me. But she said she'd never seen it anywhere else, and as far as she was concerned, it should properly be called [Craig]'s Method, and I'd be within my rights to call it that.

So there you have it - nothing is officially 'written in stone', or anything like that, but if anybody wants to make reference to the Best-Fit Circle formula by that name, I welcome you to do so. . . ;)