Generating (almost) equally-spaced points along a parabola

That's a fair critique, although I think you could mitigate that by decreasing the desired arclength - reducing Del S to 0.5 from 0.85 in your example mostly fixes the problem.

In the context of the game described in the post, the main variable we were concerned about was u_x, because we scaled difficulty by increasing it over time. This in fact results in flatter parabolas, so I actually had quite the opposite concern of what you outlined and might have over-optimized in that direction!

Thanks for taking the time to read and try out what's described in the post. It's my first time working on a technical blog, and I'm extremely grateful for any constructive feedback / criticism :)