Search index in Haskell, estimating distances with milliradians
We build a search index in 150 lines of Haskell (not much!) and use clever properties of circles to estimate distances. Also you should subscribe to the church of the radian.
There's been a lot of Haskell articles lately. If that's not your $container of $beverage, I apologise. It will not always be this way! If you like Haskell though, you're in for a good one this week.
Either way, the flashcard this week almost turned into a short article, so enjoy that too.
New articles
Search Index in 150 Lines of Haskell
We're recreating the "search index in 150 lines of Python" except in Haskell. Our code also has fewer bugs, I think. The essence of the process will be constructing millions of mini-indexes each covering just one document, and then smushing them together into a big index. That's what the semigroup pattern is about, and it's honking awesome!
Full article (11–24 minute read): Search Index in 150 Lines of Haskell
Flashcard of the week
The angular distance between lampposts on a bridge subtend 1.5 outstretched knuckles in your field of view. You estimate that you could fit 6 cars between the lampposts. How far away is the bridge?
This is a useful technique for estimating distances. It's not quite as arithmetically easy as some others (e.g. using eye parallax) but it's more flexible because it can be done non-horizontally and it teaches basic relationships of the circle. Either you know this technique or you don't, so I'm just going to work through it with you here and then we'll see the answer.
First, picture yourself at the centre of a biiig circle, whose edge intersects the two lampposts. We are asking about the distance to the lampposts. That happens to also be the radius of that imaginary circle.
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We have used our knuckles to measure the angle of the space between the lampposts. The first knuckles on your outstretched hand subtend roughly 30 milliradians. Thus, 1.5 knuckles correspond to 45 milliradians.
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The six car lengths correspond to a tangent of about 30 m. Since this is a tiny tangent (only 45 milliradians), we can assume that the corresponding circular arc is the same length.
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If 30 m is 45 milliradians in our imaginary circle, then, 2/3 metres is one milliradian.
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The radian is the amount of circular arc that equals the radius. Thus, the milliradian is the amount of circular arc that equals a thousandth of the radius.
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Wait, that's all we need to know! Our circle has a radius such that 1/1000 radii is 2/3 metres. What is the radius, i.e. distance to the bridge then?
30 m/45 mrad = 2/3 km. The bridge is ~700 m away.
Once I learned how this works, I became a convert to radians. I didn't understand why people insisted on radians when degrees are so neat, but I get it now. I wish we would just switch to milliradians everywhere. Yeah, yeah, I know it's difficult to remember a circle being 6283.18531 mrad, but that is not a practical problem. If we need precision, we can either use the computer (which knows the numeric value with sufficient precision) or maintain the calculations in symbolic terms, where the circle is exactly 2000π. Half a circle is exactly 1000π. When we divide the quarter circle into thirds (as we commonly do with compass directions), each is 500π/3. This is cumbersome and unintuitive, but if we absolutely need precision in calculations by hand, we can do it.
For the cases where we don't have a computer and don't need that level of precision, we can use a different unit, "mils", defined such that a circle is 6400 mils. This is nearly as easy to work with as degrees, and significantly more intuitive/useful. Half a circle is 3200 mils. We would no longer divide the quarter circle into thirds, but into fourths. Each 1/16th of the circle is 400 mils. So nice. A mil is nearly the same size as an mrad, so in the practical cases where we do this, we can use them interchangeably, i.e. pretend we are actually using mrads.
(In case you're curious about that eye parallax remark, here's how it works: Close one eye, then hold up a thumb on an outstretched hand, such that the thumb's edge aligns with one of the lampposts on the bridge. Without moving the thumb, switch which eye you're looking through. Estimate the distance your thumb edge moved against the background. In this case, it would be around 1.5 lamp posts, i.e. 70 metres. Multiply by 10. That's the distance.
This works because the ratio between the length of your arm and the distance between your eyes is in the region of 10:1. It's arithmetically easier because once you have estimated the distance your thumb moved, you just need to add a zero to that number to get a distance. The drawback is the line of your eyes need to be aligned with the reference distance. The knuckle method works also using e.g. standing people as references without tilting your head sideways.)
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