How to place $n$ points on a unit hemisphere, such that balls of some radius $r$ around those points cover the maximum amount of volume that lies in the region $(x, y, z)$ with $x \geq 0$, $y, z \in \mathcal{R}$, where the balls can't overlap?
First off, I should say that I'm looking for a solution that is good enough, not necessarily perfect. My intended goal is constructing a filter bank of $n$ filters with Fourier support centered at the points on the hemisphere and bandwidth $r$.
I basically see two problems that need to be overcome. Most efficient placements (such as the Fibonacci lattice
- Cover the whole sphere, not just a hemisphere.
- Maximize the area each point represents, not its volume.
As far as I can see, adapting the Fibonacci lattice to cover a hemisphere is my best bet, and it will probably involve cutting of the spiral somewhere a distance $r$ above the $(y, z)$-plane, since I really don't care about the frequencies $(x, y, z)$ with $ x < 0$.
I am really curious to hear your ideas.