I am interested in gears. Sadly most of the writing on this is very practical and does not get into abstract theory. I have been trying to formalize these ideas to be able to ask what shapes can function as gears and at what ratios. I have had two attempts to formalize this question.
For my first formalization we can think of the functions $f:[0,2\pi) \to (0,\infty)$ as defining the gears outlined by the polar expression $r=f(\theta)$. My question becomes for what values $a\in [0,\infty)$ is there a function $g:[0,2\pi) \to (0,\infty)$ that generates a gear that meshes with our first gear and rotates $a$ times for every time the first gear rotates. I included $0$ as a possible value of $a$ to represent linear gears. This formalization falls short because it only looks at particular shapes. In this formalization it is also difficult to define what it means for two gears to mesh.
For my second and preferred formalization we think of a gear as an ordered pair $G = (X,c)$ where $X$ is a compact subset of $\mathbb{R}^2$ and $c \in \mathbb{R}^2$. This is meant to represent the gear made of the points in $X$ that rotates around the point $c$. The set $X$ is not necessarily connected and $c$ is not necessarily in $X$ because of objects like lantern gears. There are a lot of ways to formalize a rotation. I chose to use elements of $S^1$ and apply them to the gear object. Now given the gears $G_1$ and $G_2$ their meshing space is $\{(a,b) \in S^1 \times S^1\mathrel\vert aG_1\cap bG_2 = \emptyset\}$. It is interesting to note that $S^1 \times S^1$ is the torus. So the question from before becomes whether or not there are continuous functions from $S^1$ to this subset of the torus, and if so how many and members and of what homotopy classes.
Edit: There are a lot of questions that could be asked about the meshing space. One could be dimensionality. If the space is made of one-dimensional curves that means the gears line up perfectly. If the space is for example the whole torus that means the gears do not touch each other at all. Also upon further playing around, I think it makes more sense to have $X$ be open and bounded. If $X$ is closed then by the intersection condition the gears could not touch each other. This also removes situations with single disconnected points.
I was wondering if anyone had an answer to my question, under either formalization, or had a smarter way to formalize this question.
