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.
The image is not mine. It is a fantastic creation bybigblueboo that has caught some attention outside of the usual math tumblverse. You should definitely check out eir blog and if you like this post you should (also?) reblog the original. With that out of the way:
Modeling. Mathematical modeling is the art of translating real systems, often physical or economic, into the language of mathematics in an attempt to predict future behavior of the system. Doing this often requires making many simplifying assumptions which are “unwarranted” from a purely logical perspective, but make the problems tractable. Therefore mathematical modeling is a distinct (but related) skill from the modern conception of the practice of mathematics.
A “parametric equation” is difficult to define exactly, but it is often (as it is here) a method for producing general surfaces or curves that cannot be described by functions because they do not pass the vertical line test. More specifically for this example, it is a function from a line segment into a higher-dimensional space.
This object has inspired me to get off my lazy butt and start producing content for the blog again. It also happens to be an excellent source of mathematical content: I’m probably going to be doing a daily series of posts about it for a while. I know I have content for at least three days and probably a few more besides.
It is vaguely related to epicycles, but the name “generalized epicycle” is my own invention. If you know an actually recognized name, I would love to know about it!
(EDIT: There is an old version of this post in which the constant terms were omitted and there were some flopped sin/cos symbols. I’m sorry! These haven’t been edited by someone else unlike many of the proofs I post so they’re bound to be a little rough around the edges every so often.)