Helical coordinate on a sphere

I set out to find the equations describing a rope that begins from the north pole and spirals its way around a sphere neatly covering it and ending at the south pole. Taking uniform samples along this rope gives a fairly uniform covering of the sphere. The surface of a sphere is two dimensional, but the rope gives a one dimensional covering that is uniformly sampled in that dimension. I wondered about the sampling in the other dimension so I set out to plot it. I found the ``generalized spiral set'' of Saff and Kuijlaars1997. In spherical coordinates $(\theta,\phi)$, for $0\le\theta\le\pi, 0\le \phi\le 2\pi$, they set

$$\begin{eqnarray} \theta_k &=& \arccos(h_k), \quad h_k \ =\ -1+ {2(k-1)\over(N-1... ...1\over\sqrt{1-h_k^2}}, \quad 2\le k\le N-1, \quad \phi_1=\phi_N=0\end{eqnarray}$$ (1) (2)

My plot of these equations is shown in Figure 1. There is no interesting pattern in the crossline direction. Although my plot looks reasonable, Saff and Kuijlaars1997 show a curious pattern in the crossline direction that my plots do not show. A few tests with various values of N and various rotations failed to show any curious pattern.

 

sphere
Figure 1 Helix on a sphere. Top shows the embedded helix. Bottom hides it. An interesting pattern of points that appears in the article in the Mathematical Intelligencer is inexplicably absent here (even though I tested several rotations and several values of N).