If I had a sphere and an octant like this, how would I find point $P$?
I can already calculate the other portions of the sphere using the parametric form equations, but I am unsure how I would combine to arrive at the $(x, y, z)$ of $P$ in the image.
For clarification in the image:
$O$ is the center of the sphere; it is not necessarily the origin of the space
Points $A, B, C, D, E, F$ are already known
Point $P$ is a point on the surface of the octant
I would've though that $P$ would've been the center of the arc from $B$ to $F$ in a similar fashion that $C$ is from $E$ to $F$.
A close approximation of this is to take the inital center point $(x, y, z)$ and make $P = (x + \epsilon/\sqrt{3}, y + \epsilon/\sqrt{3}, z + \epsilon/\sqrt{3}$) though I believe this to be incorrect.
As far as I can tell, you're asking for a point on the sphere.
By symmetry $x = y = z$. And given that $x^2 + y^2 + z^2 = e = 3 x^2$, we immediately find that $x = y = z = \sqrt{e/3}$.