Peculiar maths and periscopes.

One of the things I’ve been up to recently was making something that can take photos underwater. I figured a periscope-type affair was what I needed to make. So I looked at getting some clear plastic pipe and mirrors.

Clear plastic pipe isn’t too hard to find, as it turns out. It’s just bloody expensive when you’re prototyping stuff. What I wanted was something I could fit a 50mm lens into – something with an inner diameter of > the lens diameter (which is about 52mm) – and long enough to get under the surface of the water without getting my camera soggy. So I went to Scraps & found a 2l measuring cylinder instead. Advantages: the bottom is already sealed, so you don’t need to worry about waterproofing. It’s clear (or will be after cleaning). It is the right diameter to stuff a camera lens in. Disadvantages: it’s filthy. And will need weighting. And it’s a bit too long, so the mirror will need to be lifted off the bottom.

(This is what shooting down the measuring cylinder looked like before cleaning it, then stuffing it with sand and a broken mirror)
Messing around on boats

Anyway, I cleaned it (washing up liquid, followed by meths), put sand into some freezer bags and stuffed them in (after initially pouring the sand straight in, which of course rendered all my hard cleaning work a waste of time), then broke a bathroom mirror trying to cut it to an approximate size, and dropped it in at an approximate 45° angle.

(This is a plant in my garden, as shot through the mirror)
Messing around on boats

So, it works. On dry land. Unfortunately where I was going to test it out – the Leeds-Liverpool canal at Rodley as part of an Exposure Leeds outing with I Love West Leeds – the water was filthy and far too murky to try out. So I’ll take it to Scarborough instead, at some point. Also, I was a bit late and had to run a mile or so up the canal towpath to catch up with the boats, carrying a big glass tube, camera bag and my usual satchel with me. And it was a bit bloody warm, so to be unable to use the prototype periscope was a bit miffing.

Anyway, there was someone there who works with glass, who offered to cut me the right-size mirror to fit the tube properly instead of my destroying shaving mirrors willy-nilly making things that are potential deathtraps. But I had to work out how to make that oval first.

On the surface of things this is a pretty straightforwards problem. You need to know the angle of the mirror and the diameter of the tube it has to go down. Ok, and… then what? Ellipses are calculated on two directional radii, r1 and r2 – the area of an ellipse = π(r1r2) – but I know that one of those radii must be half the diameter of the cylinder. For the sake of argument (and maths) we’ll say the diameter is 100mm, so the short axis has r1 = 50mm. Now we need to know what the long axis of the ellipse (r2)should be. In three dimensions this is quite tricky, but you don’t need to do that, because all you need is the length of one line.

Instead, turn it into a triangle problem. You need to know the length of the hypotenuse in a right-angled triangle where the angle of inclination is 45°. As we should all be able to recite from primary school “the square of the hypotenuse is equal to the sum of the squares of the other two sides”. Both α & β are 45°, and we know the diameter (a) is 100mm so the height (b) must also be 100mm, ergo the length of c must be √2(1002), or ~141mm. So I would need an oval mirror 100mm x 140mm, which will fit nicely and shouldn’t wedge.

I’d not thought about this sort of problem for 20 years or so, so I’m quite pleased with how it came out. It’s the same if you ever need to work out the surface area of your tea when you’re tipping your mug to pour into your mouth. Tea goes colder quicker if you tilt it, because the surface of the tea is much bigger than when it’s just sat upright in a mug (and greater surface area = less heat retention, which is why heat sinks work).

I’m straying into fluid dynamics territory now, so I’ll quit while I’m ahead.

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6 Responses to Peculiar maths and periscopes.

  1. starkwe says:

    I love mixing math and art! Wonderful post.

  2. Hester says:

    Does it need to be rigid? I wonder whether plastic drinks bottles might be cheaper alternatives for prototyping, if you can find one that’s long enough to have a top cut off, and not too wide? If the mirror is cut to size, and with the sand in the bottom, it might work!

    • backwardslion says:

      I didn’t want water pressure to deform the plastic. It doesn’t take much! But it is a good idea – maybe a 1l tonic bottle might do the job, if I can clean it without scratching… Thanks!

  3. Pingback: Periscope, results section. | Backwards Lion

  4. Pingback: Messing about in boats, 2013 edition | Backwards Lion

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