[dmd]

As an RA, before starting grad school, I hacked together some code to choose a random point on a sphere, for some psychophysics experiment I was doing.

Fortunately, I never used it for anything, because I made the classic naive mistake of simply choosing a random theta in 0,2pi and phi in -pi,pi, which ends up with points biased towards the poles.

Somehow *12 years later* my subconscious flagged it up and I woke up in the middle of the night realizing the issue. Even though I'd never revisited it since then!

https://github.com/dmd/thesis/commit/bff319690188a62a79821aa...

[marcodiego]

A friend of mine made this mistake a long time ago when sampling points in a circle. He was baffled and couldn't understand why there was concentration of points around some areas of the circle. My explanation: draw a square around this circle; trace a horizontal line passing along its center; now, draw a diagonal line; can you see it is more likely that a randomly sampled point is closer to the diagonal than to horizontal line? He said "sure!" and I asked "why?" to what he promptly answered "because its longer.". He then asked what he should do about it, I simply said: "just ignore the points that are too far". He immediately understood that points inside the square but outside the disk were the reason for the concentration of the points.

[xanderlewis]

> simply choosing a random theta in 0,2pi and phi in -pi,pi, which ends up with points biased towards the poles.

Am I right in thinking that it’s because circles of constant theta on the sphere contain the same ‘expected’ number of points (since phi is uniformly distributed), and these circles get smaller (and in the limit, shrink to a point) as one travels towards the poles — hence, higher density of points…?

…makes me realise that spherical polar coordinates, as natural as they seem, aren’t in some way homogeneous in the sense that whilst the mapping (theta, phi) —> (x, y, z) is continuous, it isn’t uniformly so… or something like that. I don’t know enough (possibly differential) geometry to articulate it properly, but it feels like the problem of getting genuinely uniform sampling from a sphere (or indeed any manifold) is somewhat equivalent to having a ‘nice’ coordinate system (one that respects the symmetry of the sphere). Basically, polar coordinates are prejudiced and treat points differently depending on how close to the poles they are.

By definition, our sampling in the domain space (two intervals) is uniform; the problem comes when we project through a coordinate system that doesn’t respect this.

Which better solution did you use? I’m having trouble reading your code on my current device.

[eli_gottlieb]

> whilst the mapping (theta, phi) —> (x, y, z) is continuous, it isn’t uniformly so… or something like that.

The term is "measure preserving". The mapping is continuous but it doesn't preserve the length of intervals projected through it.

[xanderlewis]

Yes. That seems close to what I was looking for. Thanks.

[dmd]

> Which better solution did you use?

I didn't, since I haven't touched or needed this code since 2003. I just put a warning saying don't use it.