The image is not mine. It is a fantastic creation by bigblueboo 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.)