The arc-length parametrization of an Archimedean spiral visualized with colors.

I just finished some new code to generate arc-length parametrizations of arbitrary curves. To try it out, I used it on this Archimedean spiral.

Then I got curious: exactly how does the re-parametrization redistributes the points along this curve? In the original parametrization, the points are bunched up in the middle of the spiral, and more spaced on the outside. The arc-length parametrization makes them equally spaced along the whole path. So how do they compare?

First, I tried this with black points, but it was too confusing. Same thing for a few dots highlighted. So I decided to color them all based on the angle in the original parametrization. This is the result.

It is really interesting how the colors are bent around. It seems that the distribution is quite non-uniform, even though the spiral is rather uniform in growth.

I originally rendered this with four times as many frames, but due to the amount of colors and dimensions of the GIF, Tumblr wouldn’t accept it. It was too large. Below is the animation with twice as many frames.


Hint: try squinting! It blurs the colors and it looks really trippy!