1920s Radial Planimeter Review / HowTo

A K+E Radial Planimeter, from 1920s. It measures the average radius of a roughly circular shape!
This is episode 76 of my video series about old calculating devices.
Radial planimeter instructions: archive.org/de...
End song inspired by "Hotter than a Molotov" by The Coup.
Chris Staecker webarea: faculty.fairfi...
#planimetria

[@jero37]

The circular graph paper for the temperature device is such an aesthetic. Looking forward to that device's video.

[@orbatos]

I used the thermometers a lot on jobs with environmental controls. They were treated as both backup and a convenient history.

[@MarkEichin]

In the late 70s, I saw a circular chart like that still in active use in a factory - measuring temperature and humidity on the same page, as a process record (someone would change the charts daily and file them.) At that point, I don't think anyone was doing analysis on the record, it was just part of a lingering process, but it was a wonderfully intricate piece of machinery. Looking forward to the sequel!

[@Nashvillain10SE]

Soooo excited waiting for the sequel!!!! 📈

[@hughobyrne2588]

Funny how a 'radial planimeter' and 'polar planimeter' seem to be so closely related, in their names (anything but cartesian!), and even in form - this looks like it might be a modification of a polar planimeter, they just figured out a way to change one degree of freedom from a rotation to a translation - but the polar planimeter needs its anchor point to be outside the shape, and the radial planimeter needs its anchor inside. How similar they are, how dissimilar they are - it feels like there might be an interesting insight to be gained from a deeper 'compare and contrast'.

[@charleslambert3368]

i think the difference is that with a polar planimeter, rotating around an inch of curve circumference moves the wheel more, the further the wheel is from the centre. whereas with this planimeter (mean radius finder), it's the same no matter how far out you are. so one is doing ∫ r^2 dθ and the other ∫ r dθ

[@kaisalmon1646]

a sequel?! Chris Staecker cinematic universe here we goooooo!!

[@chrtrptnt]

I have 6 deep freezers at work that use those temperature logs. We have to record, save and file these for years. Though it is rare, I can be asked to provide the storage conditions of specific lots of drugs/samples for the entire time we had them. Still alive in the pharmaceutical world, even if its a backup to digital monitors.

[@ChrisStaecker]

Are they built in to the freezer? Or standalone things that you stick in there? Fully mechanical or electric?

[@chrtrptnt]

You know, I've never really investigated them now that I think about. I will check them out on Monday as now I am curious. They are built into the freezers though and we just upgraded a few years ago during covid vaccine trials so these aren't ancient machines.@@ChrisStaecker

[@yetanotherdan]

You blew my mind about the average radius depending on where the "center" is located. I was expecting you to double check by bouncing around.

[@ChrisStaecker]

Yes this was a surprise to me. I actually filmed myself doing it at different origin points, expecting to get the same answer every time- but I didn't! I thought I was doing it wrong somehow but the answers were consistent after repeated measurements, which made me eventually realize what was going on.

[@Eldriitch]

@@ChrisStaecker Yep! This is even true for finitely many points -- though the average radius is then minimized by the geometric median, which is unique for points in general position. Unfortunately the geometric median can be very hard to compute, even for finitely many points. You generally have to use some sort of iterative method which approaches it up to the desired position.

[@ChrisStaecker]

@@Eldriitch This is a good bit of intuition- I hadn't considered finitely many points. Obviously for two points the average radius depends strongly on the "center" point that you measure from, and there is a unique point which minimizes this radius. So it is not surprising that the same is true for any finite number of points, and then not too surprising that it is still true for infinitely many points. Thanks! I also didn't know the term "geometric median"- can this point be measured mechanically? Here is an idea: attach a spring to each point of your set and join all the springs together at some "center" point which is free to move. Then the springs will equalize so that their center point is the geometric median. Is that right?

[@Eldriitch]

@@ChrisStaecker There may well be a way to measure it mechanically, but I don't think the springs method would work. The equilibrium of that is where the sum over x_i - x = 0, which pretty cleanly goes to x being the mean of the x_i. In other words, their center of mass (if they all have equal unit mass). The characterising property of the geometric median is instead the point for which the sum of directions to each x_i is 0 instead. To mechanically compute this you'd need a way to apply a fixed 1 N force to the freely moving point. It's not immediately obvious to me what kind of mechanism could do that, but it also sounds like something the engineers would have some use for, so it probably exists.

[@ChrisStaecker]

@@Eldriitch Thanks- I think I am having trouble distinguishing the geometric median from the center of mass. (thanks for setting me straight!) Is what you said about directions only true for more than 2 points? If I have 2 points, there are many "center" points which make the sum of directions 0.

[@haptot]

0.0004 would be the conversion (approximately) between 0.01 mm to 1 in. The conversion factor is actually a bit smaller but I don't think that would be significant to the accuracy of this device

[@ChrisStaecker]

But there needs to be a factor of 2pi in there also, since the wheel is rolling out a circumpherence.

[@haptot]

@@ChrisStaecker I think the scales on the device have most likely accounted for 2 pi already. The factor they say to multiply by is a common approximation to use when precision to that degree is not necessary or achievable (the real conversion factor is 0.000393701).

[@ChrisStaecker]

@@haptot You are 100% right- I just measured the wheel- it is exactly 1cm radius. So the scale on the wheel is in units of mm*2pi, so conversion to inches doesn't need to divide by 2pi. Thanks for the knowledge-

[@Pallethands]

A cliff hanger? This is why I subscribe. Don't wanna miss that next video

[@supergub]

It's great to see the Pupcake training aid again!

[@ChrisStaecker]

How could I discuss a planimeter without him? (Hope he doesn't mind being stabbed in the face)

[@supergub]

Learning comes through pain - Aeschylus @@ChrisStaecker

Thanks for your videos, they're great

[@orbatos]

These plotting thermometers were still common in existing machine rooms in 2005, some forestry work and I'm sure other things as well.

[@cmdrredhawk]

Radio wave propagation charts look circular like that too. It would determine the average signal emission of an antenna.

[@haramanggapuja]

Thought the same thing. Not the Smith chart, though. Thems is inscrutable, thems is.

[@ericgarner2559]

So, believe it or not, round temperature chart recorders are still a thing in some places, especially for applications where you need a record of temperature control for audit purposes. they are still available as an add in option on some kinds of temperature chambers and scientific/industrial freezers.

[@ChrisStaecker]

Yes I've heard they're used by old timey brewers to monitor temperature and pressure. It's the kind of thing that I can imagine working for decades, and if you've got an old one it's cheap and easy to keep on using it.

[@chevanc]

I've worked in bio labs built in the mid-teens that had those kind of plotters built into the door of temperature controlled rooms. Nobody ever used them, though. Not sure we even had the papers for them.

[@larryscott3982]

You would love survey instruments of the 50s and 60s. Tachymetry was a trip.

[@michaeldabate6358]

Circular charts are still used in hydrostatic testing of welded assemblies! Most shops have a Barton 202E or equivalent. Pressure vessels and high pressure natural gas lines for weld testing primarily, I saw a few for gas flow totalizing in the field, but gas measurement is mostly digital now.

[@dimitar4y]

i absolutely love the "these thangs exist and it duuuumb" attitude.

[@seonbin2]

Warning : A modern semiconductor-based calculator appears at 4:08.

[@Saito232005]

Wow. I wish I had all these old toys. I love old instruments and tools. Much more dependable.

[@btg837]

At a large aerospace manufacturer in Missouri, circular graphs and auto-recorders are used to record temperature and atmospheric pressure in a tool-making shop. Temperature is important when measuring accurate things (as stated elsewhere in the comments). The paper wheels are archived. Thanks for the fun videos!

[@ChrisStaecker]

Do use purely mechanical ones? Or are they electric? Why not do it all digitally? "aint-broke-dont-fix-it?"

[@btg837]

@@ChrisStaecker I've got some questions out to the quality guys and will report back. They are electric and they ain't broke!

[@jamesonhardy2126]

Oh boy, can't wait.

[@vvalerio77]

A cliffhanger!

[@markgreco1962]

I like your delivery

[@someonespadre]

I found a polar planimeter in the unloved old tools drawer that looks like the one in this video, except it is the area one. It works. I didn’t bother with the vernier.

[@dangevad]

OMG. Finally. I work in a high school, I found one of these years ago but completely missing any identifying labels or instructions (and missing the spiked wheel part) and I could never work out what the hell it was.

[@ChrisStaecker]

Nice to hear they still exist in the wild! I’d never heard of it until I got mine.

[@Vallee152]

is there a video somewhere explaining how measuring from an off-centre point in a circle can increase/decrease the average radius?

[@ChrisStaecker]

No video but here is a discussion with some fairly intuitive answers. math.stackexchange.com/questions/4764228/average-distance-from-point-to-unit-circle/4764245#4764245 The average as measured from the center is the minimum possible, just giving the radius of the circle. All other off-center points will give greater answers for the average.

[@ElectroNeutrino]

Start with a parametric equation for a circle: x(t) = R cos(t), y(t) = R sin(t). The distance r from point (d,0) to any point on the circle is given by r(t)^2 = (x(t)-d)^2 + y(t)^2. The average is given by integrating r(t)^2 from 0 to 2 pi with respect to t and dividing by 2 pi. When you do that, you find that the average distance is sqrt(R^2 + d^2). Edit: D'oh. Can't just integrate r^2, and integrating over r gives an elliptic integral.

[@ChrisStaecker]

I think you need to be integrating r(t), not r(t)^2. This results in a messy elliptic integral, though you can numerically evaluate it no problem. See this: math.stackexchange.com/questions/2020713/average-distance-from-point-to-circle

[@ElectroNeutrino]

@@ChrisStaecker Yea, I caught that right after I posted. Thanks for the correction.

[@ChrisStaecker]

@@ElectroNeutrino Sorry- happens to me all the time.

[@Beldraen]

Will the sequal be better? Probably need more explosions. ;)

[@ChrisStaecker]

for some reason there's a kid

[@hughobyrne2588]

​@@ChrisStaecker ... I hope that is to say, *instead* of an explosion, and not... *along with* an explosion.

[@okancanarslan3730]

amazing odd device to quantify the outputs of another odd device

[@randomsandwichian]

You know what they say, any day is a good day with a planimenimelimeter

[@Bhatakti_Hawas]

Its a wonderful instrument

[@joshuabrigden4820]

7:00 Initially I assumed that like the function of a quadratic breaking down if 2 X points occupy the same y axis, this tool would lose accuracy if not be useless once the wheel rolled backwards. However after a little ponder, assume a max radii circle is drawn around the shape, letting the wheel roll backwards would be reducing the value due to the empty space between the max circle and the shape. There are lesser radii points along the circumference of the dog from cheek to ear, you couldn't ignore the line and got check to ear and so the rolling back of let's the tool take that into account. Yeah I thought about this too deep lol

[@jackoliver2495]

I love the Hello Kitty calculator, can't believe they came up with that in the 1920s, truly a marvel of engineering. Btw, would you ever consider doing a video on proof assistants if you've ever used them (e.g. Lean/Coq)?

[@ChrisStaecker]

I've used coq and tried lean once- I would like to learn them more seriously someday. Probably not soon though!

[@Eldriitch]

@@ChrisStaecker It can be a lot of fun. We even have a formal mathematics class here at Imperial based in Lean -- for which you can blame Kevin Buzzard naturally.

[@lourias]

Coolness!

[@retrogiftsuk4812]

Arg! The definition of planimeter is "an instrument for mechanically measuring the area of a plane figure.". Why are they using the name for this gadget? It'll encourage people to assume that you can use the average radius to calculate the area. Great video as always, and that velvet lined box takes "original packaging" to a new level. Can't wait for the sequel.

[@hughobyrne2588]

But... the pen appears to travel along an arc as the temperature changes, it's not moving directly radially towards and away from the center of rotation of the circular graph paper. The inaccuracies! Won't someone think of the first-order perturbation effects!

[@ChrisStaecker]

If you look carefully the graph paper in there takes this into account- it's not a pure polar coordinate chart. The "radial" lines have a slight arc to them. But maybe you're suggesting that the radial planimeter will no longer give an accurate reading of average radius? This might be true- I'm not sure. (obvs the effect is very very small)

[@hughobyrne2588]

@@ChrisStaecker Yeah, I saw the arcs on the graph paper - I was referring to the effect this would have on the planimeter. Maybe if the characteristics of the arc motion of the pen were figured out in the right way, you could make a planimeter where the groove on the bottom, where the pin slides, is a curve instead of a straight line? Then, the direction the roller is rolling is not (always) exactly perpendicular to the radius of the circle described by turning the arm around the pivot. Hmmm. It might work.

[@ChrisStaecker]

The planimeter calculates the average radius assuming a "straight" polar coordinate grid. So maybe it's as simple as: you measure the average "straight" radius using the planimeter, and this reading will be slightly high because the chart radii are curved. But you can measure out your "true" radius on the chart, and thereby convert it to the appropriate chart reading.

[@hughobyrne2588]

​@@ChrisStaecker Close, but not quite. At least, not if the angle of the pen directly corresponds to the function value. Imagine the pen is 10 degrees to the left for half the day, and straight up the other half. The circle corresponding to the pen at 5 degrees will not be quite the same as the circle with its radius halfway between. The dr/dy isn't constant, if that makes sense. Not trying to be difficult, I know it's small-order effects, but I find the details can be fun to track down.

[@ChrisStaecker]

@@hughobyrne2588 Yes I think you're probably right. I find this thing deceptively simple- it's clear how and why it works, but the quantity that it's measuring is pretty strange. I was confused for a while about the point I raised at 6:17- intuitively I had expected these things to be the same. Throw in a curved chart and all my intuition is out the window. Probably the correct intuition is just: it mostly works fine but things are very slightly weird.

[@charleslambert3368]

i suppose the planimeter and the radial graph plotter are like a mechanical disc integrator but split in two. As in there were devices where the pen of the plotter was replaced by a wheel (usually a ball on an axle) like on the planimeter

[@ChrisStaecker]

I'd love to get one of these- haven't found a good one yet

[@herzogsbuick]

thought maybe later, ya know, we could go out for some tapas and upscale graph paper, if, you wanted, be...be pretty coooool

[@r0bhumm]

I wonder if you could use this device with a tachograph disc I know they store speed one trace I’m not sure what the others are.

[@ChrisStaecker]

Very interesting- I'd never heard of a tachograph. Seems to be the same idea as my temperature sensor, though the picture at wikipedia has very volatile movement on the curve- it would be a pain to trace. But it should work!

[@r0bhumm]

@@ChrisStaecker tachograph were the first thing I thought of when you mentioned, circular graphs. I know they are now almost all digital devices and don’t use the specialist paper discs but when I was growing up you did hear about people fiddling them from time to time. My father used to work in the Met office, and now they had clockwork devices that recorded pin traces of both the temperature and pressure these used a drum with a paper graph round it, not a paper disc.

[@AndyLundell]

Weird that they didn't just re-calibrate the scale so the numbers were pre-multiplied. I mean, maybe dividing by a thousand would need complicated gearing, but would it have been so hard to at least pre-multiply by four?

[@ChrisStaecker]

I had the same thought- I guess they didn’t want to make the wheel bigger?

[@AndyLundell]

Oh, I see. Someone else explained it. It's calibrated in metric, and instead of explaining that to the American audience, they just say "Multiply by this weird number, and don't ask any questions."

[@Minihood31770]

-pscale graph paper

[@ChrisStaecker]

IYKYK

[@duccie]

i wonder what value youd get if you use it on a square

[@ChrisStaecker]

For the unit square, seems that the answer would be around 0.561. See math.stackexchange.com/questions/3700367/average-distance-from-a-squares-perimeter-to-its-center I actually planned on doing this in the video, but I cut it out. It's actually hard to trace a straight line with this thing, because the shape of it wants to follow circular arcs- it turns angle very easily but there is some friction when you are sliding it along the grove. To make it move in a straight line you have to move the angle and slide it at the same time and it's very hard to follow the line.

[@Rhynome]

​@@ChrisStaeckera man of commitment!

[@1906Farnsworth]

The Wikipedia article explains a lot about planimeters(probably more then you want): en.wikipedia.org/wiki/Planimeter

[@nickmoniker]

Cliffhanger!!!

[@OldManBOMBIN]

Radial Planimeter 2: Electric Watcha-madoo

[@iamjimgroth]

I feel nerdy as hell. I recognised its purpose immediately.

[@BillRicker]

Care to comment on error introduced by the circular chat recorder tinge scale ^radii^ being arcs matching the 🖊 Pen's arc of motion , vs your generated straight radii?

[@ChrisStaecker]

Yes I believe this will introduce some (very very small) error. See another comment by HughOByrne

[@BillRicker]

so @@ChrisStaecker the question remaining is whether it is truly very very small, and if the error tends to (mostly) cancel out (as what goes up must come down presuming the orbit closes*, and if there's a remaining bias (possibly if there's a hard drop/jump at end of a day/week/fortnight that did NOT close back ? *(which it of course doesn't have to, but if there's a diurnal or fortnightly cyclic signal, it will approximately close the orbit, and the examples show cases where the cyclic function is cyclic and closes nicely.) ( IF i wanted to shake the rust off my calculus I'd try to prove one or the other, but that's not where I'm leaning today )

[@joshuabrigden4820]

Can someone please explain how you read the tool. Apart from the indicator line on the right I can't see how it's meant to be read. Especially considering 3:52 you read 7 but it seems the line is directly inbetween 6 and 7. Im guessing the 5 is in line but at that distance you couldn't possibly accurately read that, and then yes the 5 is 2 increments above the 0 giving us 752_, but for the last digit that part never rolls so how could you read anything other than 0? The amount of confusion I'm having with this makes me think I'm reading it the wrong way from the beginning.

[@ChrisStaecker]

The digit of 7 is read like you say. The line really is between 7 and 8 (don't read at exactly my giant blue arrow, there is a little fixed tick mark on the right side which indicates the value). The 8 looks a bit like a 6. Then the next 2 digits are easy to see. The final digit of zero is read on the Vernier which like you said is fixed- reading the Vernier is very nonintuitive but its easy enough when you know how. Hard to explain in words but I explained it in my "caliputer" video. Also I'm sure there's many more videos you could find about verniers.

[@joshuabrigden4820]

@@ChrisStaecker thank you for the explanation 😊 I couldn't tell if I was just completely wrong or what 🤣. I love these old tools we never see anymore!

[@michaelmounts1269]

pi x radius squared will give you area….

[@ChrisStaecker]

For a circle measured from the center, yes- but for a weird shape or a circle measured off-center, pi times (average radius squared) will not equal the area.

[@timetraveller6643]

Average seismograph reading...

[@johnsrabe]

Not so! No mouse on wheel lifting a small gate that lets a baseball hit a guy on the head to remind him the game is on the radio.