Circles and triangles and chords

So. I watched this video.

And I got inspired!

Yes. This problem is hard, because it involves looking at infinitely many chords. But maybe I can gain some knowledge from looking at a subset of those chords. Maybe I can split my set of chords into smaller sets, say x sets, that all have the same size. If I can show something about one of these sets, it should go for the larger set as well.

My subset will be all the vertical chords. I can get all other sets by rotating this set in all other angles.

1) A chord is a line between 2 points on the circle. Choose 2 points randomly.

Again, I can reduce the set of all vertical chords to all chords where there’s a known distance between the 2 points. Instead of looking at all chords, where the distance can be 0 to 180 degrees, I only look at those where the distance is a multiple of, say, 10 degrees. (There will then be 9 other sets, where the distance is 1 + a multiple of 10, 2 + a multiple of 10 etc. These sets have “the same size”. This can be generalized, for any delta, not just 1.) In the following, I vary this distance.

Let’s look at 3 examples.

The chords get closer together, when they are shorter. We would expect to overestimate the number of short chords.

2) A chord is defined by its midpoint, chosen randomly within the circle. (Note: for this example I don’t start out with vertical chords.)

I construct a grid of horizontal and vertical lines and look at all points in this grid. (A point is where 2 lines cross.) (I throw points away, if they are outside the circle.) (I also only look at one quadrant.) I have reduced the number of chords in this way. All other sets can be produced by translating the original grid. (In this case the sets are approximately of the same size, not actually the same size.)

I rotate points to be on the x-axis, if they aren’t already. (Now all chords will be vertical.) Then I construct the chords.

Let’s look at 2 examples.

There’s a lot of empty space in the middle. We would expect to underestimate the number of long chords. (I haven’t checked whether some of these lines actually represent 2 or more chords.)

3) A chord is defined by its midpoint, chosen randomly on a radius.

I only look at the 2 radii, that coincides with the x-axis. That way all chords are vertical. All other sets can be produced by rotating my set of chords.

Let’s look at 3 examples.