In 2D, using polar coordinates, I have divided a unit circle into n equal parts (of equal $\theta$) and been able to form equations for the radius arc between the $\theta$ boundaries of the m'th element (m ranges from 0 to 2$\pi$ when m=n). This was easy as there was only one variable to partition i.e. $\theta$ divided into n angles of 2$\pi\div\ n$. I was also able to find the polar coordinates of the mid-point of the m'th equal arc.
Now the point in my doing this was to learn patterns to extrapolate this to a 3D equivalent, and to use 3D polar coordinates to divide a solid angle up equally into n equal parts - of size $4\pi\div\ n$
My question relates to this. One could use the $\delta\Omega =\sin\theta\delta\theta\delta\Phi$ integral to integrate over partitioned coordinates. One could assume a reference start point ($\theta=0,\Phi=0$ on Y-axis etc). But it is self evident in 2D that partitioning that $\theta$ can be equally split and the parts represent equal symmetric elements and their midpoints would be as far away from each other as possible. Now in 3D, partitioning by $\theta,\Phi$ doesn't guarantee that each surface element would have the same surface area, and the centre of each surface area would be as far away from each other as possible. I am looking for a way to generate equations in 3D polar coordinates of the m'th element of the whole solid angle split into n elements. This will come in the form of $\theta$ and $\Phi$ boundaries for each element. Just dividing both angles equally doesn't mean equal surface areas, nor are equal surface areas necessarily not produced by differentially dividing each angle, I.e. long and thin. It's the two angle property which I'm struggling to get my head around splitting in order to get symmetrical equal surface areas on the unit sphere. I also need to polar equations of their centres too!
Help would very much be appreciated.
Mark