Deep End Geometry Revisited

(Note to Nathan: It's math again, but this time with no numbers at all! Only pictures! If it still doesn't make sense, well, the pictures are pretty....)


This is the type of picture I've been making for the past three weeks. That rainbow blob thing in the middle is made up of several thousand dots¹. Wouldn't it be nice if I could stretch them out to fill the whole box instead of having them all jammed together in the middle?


Here's the direction I want to stretch them. The one on the right is a bird's eye view of the same thing. Unfortunately, I can't stretch them out that way. I can only stretch them in one of these ways:


Obviously, then, I should move the blob until it's pointing in a better direction. To be able to stretch them the way I want, it needs to be pointed like so:

See how easy that was in Photostudio? Unfortunately, it doesn't work like that when it's a bunch of dots on a grid. To do what I want, I need to either move the grid around to match the blob, or move the blob around to match the grid.

Lots of people made lots of suggestions, all of which had strange side effects, so in the end, I didn't end up using any of them. See, it turns out that although it's hard to rotate a blob of dots when you have them in a box, it's not at all hard when you've got them on a circle². In fact, it's ridiculously easy.


Yep. That's what I call thinking outside the box. :) And it allows me to do this:

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Footnotes for the advanced class:

1. 79,744 total data points. Which needed to be rotated 55 degrees counterclockwise.

2. Cartesian coordinates are X, Y, Z. Polar coordinates are directions with magnitudes (or angles with radii) (or, in 3D, it's azimuth and elevation to go with the radii). Rotating in polar is easy because all you do is add to the angle. So I converted from cartesian to polar, did some addition, and then converted from polar back to cartesian. Voila!