I have 2 points $A=(x_A,y_A), B=(x_B,y_B)$ on a unit circle $O$. The distance between $A$ and $B$ goes through the perimeter of the circle. How can I transform this space to a space with higher dimensions where the distances can be computed using Euclidean formula, and the original distances are preserved as much as possible? In fact, I don't know what is the main field of math concerning such transformations.
Your help is appreciated.
On
You have to keep in mind that Euclidean distances do not behave like arc length distances, so it is unlikely that you will be able to do this.
Notice that you will need to map the points of the circle to some closed curve. The fact that you need the distances of opposite points of the circle preserved, means for every point $\mathbf{x}$ on your closed curve you need to have a unique point at distance $\pi$ from $\mathbf{x}$. So you will be mapping your circle to some closed curve on an $n$-sphere (this is just an $n$-dimensional analog of the sphere in $n$ dimensions) having diameter $\pi$.
I'll admit I'm not sure exactly where it will break down, or what the optimal way to map it will be.
Hint:
Map the points using the transform
$$\tan(z)=\frac yx,$$ where $z$ is evaluated on four quadrants, and the distance between $A$ and $B$ turns to the Euclidean $$|z_A-z_B|.$$
Unfortunately, an essential nonlinearity remains because of phase wraparound, and the exact formula must be
$$\pi-||z_A-z_B|-\pi|.$$