Comment by Sharlin

For constant speed you need a so-called “Euclidean parameterization” where the t value is proportional to s, the Euclidean distance traveled (and thus no matter the value of t, if you add some dt it always works out to the same ds). This is super commonly needed when animating motion along all sorts of curves, as you might guess.

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!