I've been doing some reading on how hypercomplex fractals are calculated using cartesian to n-spherical coordinate conversion. I've specifically been looking at these two sources:
https://archive.bridgesmathart.org/2010/bridges2010-247.pdf
http://www.bugman123.com/Hypercomplex/
To lay it out, both sources explain how the equation for a Mandelbrot set ($z_{n+1} = z^2 + c$) can be converted away from using complex numbers and instead treat the complex number $c$ as a polar coordinate, with the real number being the $x$ value and the imaginary component being the $y$ value. This allows Mandelbrot sets to be expanded to n-dimensions by simply using n-dimensional Cartesian coordinates to n-Spherical coordinates, thus how the Mandelbulb was created.
What I'm struggling with is how these cartesian to spherical conversion equations were derived in the two referenced sources. In both sources, they used the following equations to derive spherical coordinates from caresian coordinates:
$\rho = \sqrt{x^2 + y^2 + z^2}$
$\theta = \arctan{\dfrac{y}{x}}$
$\phi = \arcsin{\dfrac{z}{\rho}}$
The first two make sense just using the generalized formula for cartesian to n-spherical coordinates shown in wikipedia here. Using the recipe listed there gives me:
$\rho = \sqrt{x^2 + y^2 + z^2}$
$\theta = \arctan{\dfrac{y}{x}}$
$\phi = \arctan{\dfrac{\sqrt{x^2 + y^2}}{z}}$
The equation for $\phi$ I calculated myself is likewise what Daniel White originally formulated in his original proposal for using spherical coordinates. Understanding that $\arctan{\dfrac{O}{A}} = \arccos{\dfrac{A}{H}} = \arcsin{\dfrac{O}{H}}$ where $H = \sqrt{O^2 + A^2}$, I arrive at this: $\phi = \arctan{\dfrac{\sqrt{x^2 + y^2}}{z}} = \arccos{\dfrac{z}{\sqrt{{\sqrt{x^2+y^2}}^2 + z^2}}} = \arccos{\dfrac{z}{\sqrt{x^2+y^2+z^2}}} = \arccos{\dfrac{z}{\rho}}$
If I were to instead to try an get $\phi$ in terms of $\arcsin$ I would get: $\phi = \arctan{\dfrac{\sqrt{x^2 + y^2}}{z}} = \arcsin{\dfrac{\sqrt{x^2 + y^2}}{\sqrt{x^2+y^2+z^2}}} = \arcsin{\dfrac{\sqrt{x^2 + y^2}}{\rho}}$
This result isn't as convenient computation-wise, but the main issue is I don't understand why the two referenced sources are showing $\phi = \arcsin{\dfrac{z}{\rho}}$ instead of $\phi = \arccos{\dfrac{z}{\rho}}$. This surely can't be a mistake or typo on their part, as the second source I listed is the original Mandelbulb creator, so I'd appreciate some understanding as to what I missed.