common period. The least value of $n$, say $n_\ell$, is determined by $$\begin{aligned}
    n_\ell &= \frac{2b}{\mathrm{GCD}\bigr[2 b ,(b-a)\bigr] } , 
\label{Nl} \end{aligned}$$ where $\mathrm{GCD}\left[x,y \right]$ is the greatest common divisor of $x,y$. The projected curve in the reduced two-dimensional space is closed if $\ell$ is rational. The closed curve consists of $n_\ell$ elements; each element starts off $\rho_{max}$, evolves through $\rho_{min}$, and ends at $\rho_{max}$. Till the curve returns to its starting point, $\varphi$ laps $2\pi m_\ell$, i.e., the curve wraps around the rotation axis $m_\ell$ times, where $m_\ell$ is given by $$\begin{aligned}
\
 m_\ell = \frac{1-\ell}{2} n_\ell. \end{aligned}$$ Suppose a closed geodesic in the reduced two-dimensional space which consists $n$ elements and which wraps around the rotation axis $m$ times. The parameter $\ell$ for this string is given by $$\begin{aligned}
    \ell = 1-\frac{2m}{n}. 
\end{aligned}$$ When $\ell$ is irrational number, the two-dimensional geodesic is not closed.

The strings with rational $\ell$ have periodic structure in $z$ with the period $Z_p=n_\ell q\sigma_p=\pi n_\ell q/\Omega$. On the other hand, the strings with irrational $\ell$ have no periodicity. We show the shapes of strings in Figs. 6 for $\ell = 1/5$, $1/3$, and $1/2$.

![Strings in the cases $\ell = 1/5, 1/3$ and 