Question about angle conventions and Hopf coordinates (Appendix B of Cohl 2013)

[ibaris]

Describe the bug

Question about angle conventions and Hopf coordinates (Appendix B of Cohl 2013)

Hi,
this is not a bug report, but I noticed that the “Question” button in the documentation currently leads to a Page not Found.

My actual question is about angle conventions for the (d=4) polyspherical coordinate systems and whether they can be applied in this package.

In the cited reference (H. S. Cohl, Fourier, Gegenbauer and Jacobi Expansions, arXiv:1209.6047), Appendix B describes five topological trees for polyspherical coordinates in (d=4).

In particular, the fifth tree (type ca²) corresponds to the Hopf coordinate system, where the coordinates are

$$x_1 = r \cos\vartheta\cos\phi_1,\quad x_2 = r \cos\vartheta\sin\phi_1, \quad x_3 = r \sin\vartheta\cos\phi_2,\quad x_4 = r \sin\vartheta\sin\phi_2, \quad$$

with angle ranges

$$\vartheta \in [0,\tfrac{\pi}{2}],\qquad \phi_1,\phi_2 \in [0,2\pi).$$

My question is:

Can these Hopf-based coordinate ranges be used directly in the ultrasphere-harmonics package?

More concretely, would it be valid to adopt the angle convention from Appendix B where

• $(\theta_1 \in [0,\pi])$,
• and the remaining two angular variables are allowed to run over $[0,2\pi)$,
  as done in the Hopf coordinate version of the polyspherical system?

I want to understand whether the library expects the standard recursive hyperspherical convention
(two angles in $([0,\pi])$, one in $[0,2\pi)$ )
or whether the Hopf-based convention is also supported or could be safely substituted.

Thanks in advance for any clarification!

To Reproduce

this is not a bug report, but I noticed that the “Question” button in the documentation currently leads to a Page not Found.

Additional context

No response

Version

1.3.0

Platform

Windows

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