The End 2016 Mathematics A To Z: Osculating Circle


I’m happy to say it’s another request today. This one’s from HowardAt58, author of the Saving School Math blog. He’s given me some great inspiration in the past.

Osculating Circle.

It’s right there in the name. Osculating. You know what that is from that one Daffy Duck cartoon where he cries out “Greetings, Gate, let’s osculate” while wearing a moustache. Daffy’s imitating somebody there, but goodness knows who. Someday the mystery drives the young you to a dictionary web site. Osculate means kiss. This doesn’t seem to explain the scene. Daffy was imitating Jerry Colonna. That meant something in 1943. You can find him on old-time radio recordings. I think he’s funny, in that 40s style.

Make the substitution. A kissing circle. Suppose it’s not some playground antic one level up from the Kissing Bandit that plagues recess yet one or two levels down what we imagine we’d do in high school. It suggests a circle that comes really close to something, that touches it a moment, and then goes off its own way.

But then touching. We know another word for that. It’s the root behind “tangent”. Tangent is a trigonometry term. But it appears in calculus too. The tangent line is a line that touches a curve at one specific point and is going in the same direction as the original curve is at that point. We like this because … well, we do. The tangent line is a good approximation of the original curve, at least at the tangent point and for some region local to that. The tangent touches the original curve, and maybe it does something else later on. What could kissing be?

The osculating circle is about approximating an interesting thing with a well-behaved thing. So are similar things with names like “osculating curve” or “osculating sphere”. We need that a lot. Interesting things are complicated. Well-behaved things are understood. We move from what we understand to what we would like to know, often, by an approximation. This is why we have tangent lines. This is why we build polynomials that approximate an interesting function. They share the original function’s value, and its derivative’s value. A polynomial approximation can share many derivatives. If the function is nice enough, and the polynomial big enough, it can be impossible to tell the difference between the polynomial and the original function.

The osculating circle, or sphere, isn’t so concerned with matching derivatives. I know, I’m as shocked as you are. Well, it matches the first and the second derivatives of the original curve. Anything past that, though, it matches only by luck. The osculating circle is instead about matching the curvature of the original curve. The curvature is what you think it would be: it’s how much a function curves. If you imagine looking closely at the original curve and an osculating circle they appear to be two arcs that come together. They must touch at one point. They might touch at others, but that’s incidental.

Osculating circles, and osculating spheres, sneak out of mathematics and into practical work. This is because we often want to work with things that are almost circles. The surface of the Earth, for example, is not a sphere. But it’s only a tiny bit off. It’s off in ways that you only notice if you are doing high-precision mapping. Or taking close measurements of things in the sky. Sometimes we do this. So we map the Earth locally as if it were a perfect sphere, with curvature exactly what its curvature is at our observation post.

Or we might be observing something moving in orbit. If the universe had only two things in it, and they were the correct two things, all orbits would be simple: they would be ellipses. They would have to be “point masses”, things that have mass without any volume. They never are. They’re always shapes. Spheres would be fine, but they’re never perfect spheres even. The slight difference between a perfect sphere and whatever the things really are affects the orbit. Or the other things in the universe tug on the orbiting things. Or the thing orbiting makes a course correction. All these things make little changes in the orbiting thing’s orbit. The actual orbit of the thing is a complicated curve. The orbit we could calculate is an osculating — well, an osculating ellipse, rather than an osculating circle. Similar idea, though. Call it an osculating orbit if you’d rather.

That osculating circles have practical uses doesn’t mean they aren’t respectable mathematics. I’ll concede they’re not used as much as polynomials or sine curves are. I suppose that’s because polynomials and sine curves have nicer derivatives than circles do. But osculating circles do turn up as ways to try solving nonlinear differential equations. We need the help. Linear differential equations anyone can solve. Nonlinear differential equations are pretty much impossible. They also turn up in signal processing, as ways to find the frequencies of a signal from a sampling of data. This, too, we would like to know.

We get the name “osculating circle” from Gottfried Wilhelm Leibniz. This might not surprise. Finding easy-to-understand shapes that approximate interesting shapes is why we have calculus. Isaac Newton described a way of making them in the Principia Mathematica. This also might not surprise. Of course they would on this subject come so close together without kissing.

Reading the Comics, November 23, 2016: Featuring A Betty Boop Cartoon Edition


I admit to padding this week’s collection of mathematically-themed comic strips. There’s just barely enough to justify my splitting this into a Sunday and a Tuesday installment. I’m including a follow-the-bouncing-ball cartoon to make up for that though. Enjoy!

Jimmy Hatlo’s Little Iodine from the 20th originally ran the 18th of September, 1955. It’s a cute enough bit riffing on realistic word problems. If the problems do reflect stuff ordinary people want to know, after all, then they’re going to be questions people in the relevant fields know how to solve. A limitation is that word problems will tend to pick numbers that make for reasonable calculations, which may be implausible for actual problems. None of the examples Iodine gives seem implausible to me, but what do I know about horses? But I do sometimes encounter problems which have the form but not content of a reasonable question, like an early 80s probability book asking about the chances of one or more defective transistors in a five-transistor radio set. (The problem surely began as one about burned-out vacuum tubes in a radio.)

Daniel Beyer’s Long Story Short for the 21st is another use of Albert Einstein as iconic for superlative first-rate genius. I’m curious how long it did take for people to casually refer to genius as Einstein. The 1930 song Kitty From Kansas City (and its 1931 Screen Songs adaptation, starring Betty Boop) mention Einstein as one of those names any non-stupid person should know. But that isn’t quite the same as being the name for a genius.

My love asked if I’d include Stephen Pastis’s Pearls Before Swine of the 22nd. It has one of the impossibly stupid crocodiles say, poorly, that he was a mathematics major. I admitted it depended how busy the week was. On a slow week I’ll include more marginal stuff.

Is it plausible that the Croc is, for all his stupidity, a mathematics major? Well, sure. Perseverance makes it possible to get any degree. And given Croc’s spent twenty years trying to eat Zebra without getting close clearly perseverance is one of his traits. But are mathematics majors bad at communication?

Certainly we get the reputation for it. Part of that must be that any specialized field — whether mathematics, rocket science, music, or pasta-making — has its own vocabulary and grammar for that vocabulary that outsiders just don’t know. If it were easy to follow it wouldn’t be something people need to be trained in. And a lay audience starts scared of mathematics in a way they’re not afraid of pasta technology; you can’t communicate with people who’ve decided they can’t hear you. And many mathematical constructs just can’t be explained in a few sentences, the way vacuum extrusion of spaghetti noodles could be. And, must be said, it’s often the case a mathematics major (or a major in a similar science or engineering-related field) has English as a second (or third) language. Even a slight accent can make someone hard to follow, and build an undeserved reputation.

The Pearls crocodiles are idiots, though. The main ones, anyway; their wives and children are normal.

Ernie Bushmiller’s Nancy Classics for the 23rd originally appeared the 23rd of November, 1949. It’s just a name-drop of mathematics, though, using it as the sort of problem that can be put on the blackboard easily. And it’s not the most important thing going on here, but I do notice Bushmiller drawing the blackboard as … er … not black. It makes the composition of the last panel easier to read, certainly. And makes the visual link between the paper in the second panel and the blackboard in the last stronger. It seems more common these days to draw a blackboard that’s black. I wonder if that’s so, or if it reflects modern technology making white-on-black-text easier to render. A Photoshop select-and-invert is instantaneous compared to what Bushmiller had to do.

Bringing Up Arthur Christmas Again


Since it’s the week for this, I would like to remind folks they could be watching the Aardman Animation film Arthur Christmas. Also, I was able to spin out a couple of mathematical and physics questions from one scene in the film. Last year I collected links to the essays — there’s five of them — into a single cover page. I hope you’ll consider them.

Reading the Comics, September 21, 2013


It must have been the summer vacation making comic strip artists take time off from mathematics-themed jokes: there’s a fresh batch of them a mere ten days after my last roundup.

John Zakour and Scott Roberts’s Maria’s Day (September 12) tells the basic “not understanding fractions” joke. I suspect that Zakour and Roberts — who’re pretty well-steeped in nerd culture, as their panel strip Working Daze shows — were summoning one of those warmly familiar old jokes. Well, Sydney Harris got away with the same punch line; why not them?

Brett Koth’s Diamond Lil (September 14) also mentions fractions, but as an example of one of those inexplicably complicated mathematics things that’ll haunt you rather than be useful or interesting or even understandable. I choose not to be offended by this insult of my preferred profession and won’t even point out that Koth totally redrew the panel three times over so it’s not a static shot of immobile talking heads.

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Reading the Comics, February 13, 2013


I will return to the plausibility of Rex Morgan, MD, considered as a statistics problem, though I need time to do that pesky thinking thing and I’ve had all sorts of unreasonable demands on my time, such as work expecting me to work, and scheduling it all is such a problem. But I’ve got a fresh batch of eight comic strips that in some way mention mathematics, and I wrote paragraphs on most of those as they turned up in the day’s pages, so I can share those with you. Also I’m pleased to see that posting a Rex Morgan strip for the purpose of talking about it hasn’t brought about the end of the world, so I may be able to resume covering King Features or North American Syndicate strips when they say things worth mentioning (which they seem to do less than the Gocomics.com comics do, but I haven’t done the statistics to make that claim seriously).

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Arthur Christmas and the End of Time


In working out my little Arthur Christmas-inspired problem, I argued that if the reindeer take some nice rational number of hours to complete one orbit of the Earth, eventually they’ll meet back up with Arthur and Grand-Santa stranded on the ground. And if the reindeer take an irrational number of hours to make one orbit, they’ll never meet again, although if they wait long enough, they’ll get pretty close together, eventually.

So far this doesn’t sound like a really thrilling result: the two parties, moving on their own paths, either meet again, or they don’t. Doesn’t sound quite like I earned the four-figure income I got from mathematics work last year. But here’s where I get to be worth it: if the reindeer and Arthur don’t meet up again, but I can accept their being very near one another, then they will get as close as I like. I only figured how long it would take for the two to get about 23 centimeters apart, but if I wanted, I could wait for them to be two centimeters apart, or two millimeters, or two angstroms if I wanted. I’d pay for this nearer miss with a longer wait. And this gives me my opening to a really stunning bit of mathematics.

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Six Minutes Off


Let me return, reindeer-like, to my problem, pretty well divorced from the movie at this point, of the stranded Arthur Christmas and Grand-Santa, stuck to wherever they happen to be on the surface of the Earth, going around the Earth’s axis of rotation every 86,164 seconds, while their reindeer and sleigh carry on orbiting the planet’s center once every \sqrt{2} hours. That’s just a touch more than every 5,091 seconds. This means, sadly, that the reindeer will never be right above Arthur again, or else the whole system of rational and irrational numbers is a shambles. Still, they might come close.

After all, one day after being stranded, Arthur and Grand-Santa will be right back to the position where they started, and the reindeer will be just finishing up their seventeenth loop around the Earth. To be more nearly exact, after 86,164 seconds the reindeer will have finished just about 16.924 laps around the planet. If Arthur and Grand-Santa just hold out for another six and a half minutes (very nearly), the reindeer will be back to their line of latitude, and they’ll just be … well, how far away from that spot depends on just where they are. Since this is my problem, I’m going to drop them just a touch north of 30 degrees north latitude, because that means they’ll be travelling a neat 400 meters per second due to the Earth’s rotation and I certainly need some nice numbers here. Any nice number. I’m putting up with a day of 86,164 seconds, for crying out loud.

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Arthur Christmas and the Least Common Multiple


I left Arthur Christmas and Grand-Santa in a hypothetical puzzle, inspired by the movie, with them stranded on a tiny island while their team of flying reindeer and sleigh carried on in a straight line without them. I am assuming for the sake of an interesting problem that this means the reindeer are carrying on the Great Circle route, favored by airplanes and satellites, and that the reindeer are in an orbit more like the satellite’s than the reindeers — that is, they keep to a circle in a plane which isn’t rotating while the Earth does, since otherwise, Arthur and Grand-Santa have to wait only for the reindeer to finish one lap around the planet and somehow get up to flying altitude to be picked up. If the reindeer aren’t rotating the with the Earth, then, when the reindeer finish one circuit our heroes are going to be … well, maybe east, maybe west, of the reindeer; the problem is, they’re going to be away.

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Returning to Arthur Christmas


As promised, since I’ve got the chance, I want to return to the question of the reindeer behavior as shown in the Aardman movie Arthur Christmas, and what would ultimately happen to them if the reindeer carry on as Grand-Santa claims they will. (Again, this does require spoiling a plot point of the film and so I tuck the rest behind a cut.)

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Could “Arthur Christmas” Happen In Real Life?


If you haven’t seen the Aardman Animation movie Arthur Christmas, first, shame on you as it’s quite fun. But also you may wish to think carefully before reading this entry, and a few I project to follow, as it takes one plot point from the film which I think has some interesting mathematical implications, reaching ultimately to the fate of the universe, if I can get a good running start. But I can’t address the question without spoiling a suspense hook, so please do consider that. And watch the film; it’s a grand one about the Santa family.

The premise — without spoiling more than the commercials did — starts with Arthur, son of the current Santa, and Grand-Santa, father of the current fellow, and a linguistic construct which perfectly fills a niche I hadn’t realized was previously vacant, going off on their own to deliver a gift accidentally not delivered to one kid. To do this they take the old sleigh, as pulled by the reindeer, and they’re off over the waters when something happens and there I cut for spoilers.

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