Cusps and boundaries of connected fundamental domains for ${\mathrm{\Gamma }}_0(N)$

For $N>1$, we constructed a canonical connected fundamental domain for ${\mathrm{\Gamma }}_0(N)$ in [Nie, Parent], utilizing an interesting function $W:\mathbb{Z}\mathrm{/}N\to \mathbb{N}$. In this paper, we further study the function $W$, prove some identities, and use it to match the cusps, with widths, produced by our connected fundamental domain with the known cusp classes of ${\mathrm{\Gamma }}_0(N)$. Furthermore, we list the boundary arcs and the gluing patterns of our connected fundamental domain, a key step in understanding the modular curve $X_0(N)$ by this approach.