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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
, for
, 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}](img3.gif) |
(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).
Next: Helical coordinate on a
Up: Claerbout: Helical meshes on
Previous: INTRODUCTION
Stanford Exploration Project
4/20/1999