The easiest way (to my humble understanding) to think about the group $\Bbb S^1$ is to consider the set of all complex numbers $z=a+bi$, for which $a^2 + b^2=1$ and use multiplicative operation to construct group of the rotations of the plane, the simplest torus $\Bbb S^1$.
I was wondering if there is any neat way to modify this approach and introduce a group structure for the orbit of the earth, which is an ellipse (if it has not recently changed).
Namely, could we put a group structure on an ellipse by some modifications of the method above? Even if it demands working in another geometry!
I know that the existence of two-square identity is a fundamental fact in this construction. So, probably one should get it fixed for the generalization, if there is any. For instance, stretching or compressing size of the distance of the points on the orbit to the closer focus is the most naive idea that may come to mind. However, looking at the fundamental geometrical property of an ellipse (the fact that sum of distances of each point to the focuses is a constant) might be slightly more interesting.
I would appreciate clear ideas and references you might know for studying about this question.
You can easily construct a bijection (of sets) between $S^1$ and an ellipse, call it $E$. Once you have this bijection $\phi: E \to S^1$ put the following group structure on the ellipse:
$$a b = \phi^{-1}(\phi(a) \phi(b))$$
where you are taking the multiplication $\phi(a) \phi(b)$ in the circle. Basically what you are doing is going from the ellipse to the circle to multiply and then coming back to the ellipse.