Consider the following model, and, in particular, the second picture given in the link, namely the blue line which represents a trajectory which I call $\Gamma$.
Is it possible to give any parametrization $h\colon S^1\to\Gamma$, where $S^1$ is the unit sphere?
I think this should be possible to find auch a bijective map. Intuitively, the sphere and the trajectorie rather look "equivalently". Maybe something like the following:
Let $\gamma\in\Gamma$ a point on the trajectory. Then $\gamma=h(s)$ where $s$ is the point on the unit sphere with "appropriate" angles.