Interesting topic! However, I would like to point out that the dichotomy closed form <-> numerical methods is somewhat artificial. Even if one could express an

Interesting topic! However, I would like to point out that the dichotomy closed form <-> numerical methods is somewhat artificial. Even if one could express an arc length parametrization using exp and log, one would still need numerical methods to compute exp and log. This somehow leads to the next question: What kind of functions are suitable to describe the arc length?