Awesome visualizations. The part that I was expecting to see but didn't: how can you move at a constant speed? For the original purpose of positioning objects along a path, it doesn't matter. But when moving, you can see it's moving much more slowly at the beginning and end (mostly determined by the radius). What if I want it to travel at a constant rate? Or even apply an easing function to the speed? I'm sure there'…
Unfortunately, there’s usually no closed-form solution for it, so we have to do it numerically. And for doing that there’s in general no better way than at each t, binary/interpolation search a dt that roughly corresponds to the ds that you want (start with the previous dt, it’s likely a very good approximation).
In practice, you’d do that once and store the results, basically approximating the curve as a polyline of evenly-spaced points– at least assuming that the curve itself isn’t changing over time!