Does there exist a meaningful definition of the angle between two points on the torus?
I am working on $\mathbb T^2 = \mathbb S^2 = \mathbb R^2 / \mathbb Z^2$.
The representation I choose is $[-\frac12,\frac12)^2$.
First I show the two points in the $[0,1)^2$ representation:
Point with angle $\alpha$ rotated by $\delta$:
Point with angle $\alpha$ rotated by $\delta$ (two black points), this time in the representation $[-\frac12,\frac12)^2$ depicted by the orange dashed square.
For my application I prefer the second representation (the absolute vector length is minimized), however then
- the initial point $v$ is shown in the bottom left square, and
- the final point $v\prime$ is shown in the upper right square.
Due to this, the angle between $v$ and $v\prime$ does depend on the chose of representation? As in the first image it was $\delta$ but in the second image it is a bit less than 180° which clearly do not coincide. Is there a meaningful definition that is independent of the torus coordinates chosen?
tl; dr: There is a well-defined notion of angle between two tangent vectors at a point on a flat torus, but the concept of "angle between two line segments" is not well-defined.
We can identify "most" of the torus with a Euclidean square. The problem is, as you've noticed, that each point of the torus corresponds not with a point of the Euclidean plane, but with a Euclidean lattice. If $\bar{p}$ and $\bar{q}$ are points of the torus, and if $p$ and $q$ are Euclidean points that project to $\bar{p}$ and $\bar{q}$ under the quotient map, the translation class of the segment from $p$ to $q$ is not well-defined. Particularly, we can fix $p$ and choose $q$ to be any of the infinitely many points in the lattice projecting to $\bar{q}$.
Unfortunately, choosing representatives that are "at minimum distance" doesn't fix the issue. For example, if $p$ lies on the horizontal midline of the unit cell, there are points $q_{1}$ and $q_{2}$ on the top and bottom edges that project to the same point of the torus. The segments $pq_{1}$ and $pq_{2}$ need not be parallel.
By contrast, in the Euclidean plane itself we have a natural correspondence between vectors (based at an arbitrary point!) and directed segments, i.e. ordered pairs of points. That's why we can be sloppy about tangent vectors versus line segments in Euclidean geometry.