Without iteration and large enough p (or test for singularity) the worst quality I get is 95%. This is already a major improvement over Nowell and brute contraction. I tried the simplest iteration I could come up with just to see if it would give further improvement (which it does).

$\lim\limits_{p \to \infty} x^p = \begin{cases}0&for\ \left\vert \text{} x\right\vert <1\\ 1&for\ \left\vert \text{} x\right\vert =1\end{cases}$ so it is a (clunky) substitute for the simple test and (at least in the limit) the quality at the central line and edges is not affected.

Yes that is correct for small p. It distracts from the central question though - can the mapping in the "interior" be improved further?

Similar to Nowell, I wanted to come up with closed form expressions of the form $\underline{x}_{sphere} =\underline{f} \left( \underline{x} \right)$. Hence the idea to include the power term to deal with the singularity. The central coordinate line and edges remain straight under the transformation $\underline{x}_{c} =\underline{g} \left( \underline{x} \right)$ and the "quality" cannot be improved. However, elsewhere the iteration helps a bit. Since I was so close to achieving 100% quality throughout, I wondered if there was an alternative (closed-form) expression that would achieve this.

Yes @HelloGoodbye. Have a look at the last figure. The map of the central coordinate line is longer than that of the edges by a factor $\frac{\pi }{4} /arctan\left( \frac{1}{2} \sqrt{2} \right)$

Thanks @user 1277628. However, the mapping I am looking for is neither area nor angle preserving. I only require equidistant points on the cube's coordinate lines to be mapped to equidistant points on the sphere's coordinate lines (as per the original query and pictures). Also see: hvlanalysis.blogspot.com/2023/05/mapping-cube-to-sphere.html

It is explained here: hvlanalysis.blogspot.com/2023/05/…

That is for the 9x9x9 grid and p=50. However, for this grid any even p>=16 will achieve better than 96.0 quality. The best I can achieve for this grid is 96.1 for p>=24. For a finer grid, a larger p is required to deal with deteriorating quality near the edges of the cube.

Sam, I want to make the grid points on a “great circle” (in the meaning discussed with Timothy) equidistant. At the moment some “great circles” are worse than others.

Thanks Daniel, I have updated the description slightly.

Hi Timothy, you are right. That’s what I meant. And the quality then becomes the uniformity of the map of those squares (or what I call coordinate lines in my response to Mark).