This is just the first post for today. There’ll be two more, so be sure to check them out later if you miss them!
Here’s an arc-length parametrization of a closed curve for the Greek lowercase letter pi, famously used for the circle constant, π = 3.1415926535897932384626… (that’s what I bothered memorizing!)
Arc-length parametrizations are also called unit-speed parametrizations, because a point moving along the path will move with speed 1: the point moves 1 unit of arc-length per 1 unit of time.
It is generally very hard, if not impossible, to find this parametrization in closed form. But it always exists for nice continuous curves. Since it has some pretty cool uses, just knowing it exists is a powerful enough tool for mathematicians to use it on other cool theorems.
Using computers, we can usually approximate it numerically to any degree of accuracy we desire. The basic algorithm is pretty simple: just make a table of arc-length for each value of t. Then, the unit parametrization is just reading the table in reverse: find t given arc-length. Some interpolation is usually necessary.