This question is a revision of the math exchange post found here.
Consider the following:
- A sphere, $S$, with radius $r_1$.
- N regions projected onto $S$, whose projections, $\left\lbrace E_i \right\rbrace_{i=1}^N$, are circles of the same radius $r_2$. Which ever is best for you to visualize, the radius could be a distance on the sphere or $\frac{1}{2}$ chord distance (diameter) going through the sphere from one end of $E_i$ to the other endpoint.
- And we also assume: $\left(\text{sum of the areas of }E_i \right) \qquad \sum_{i=1}^N \text{area}\left( E_i\right) > 4\cdot \pi \cdot r_1^2\qquad\text{(Surface Area of S)} $ and $\left\lbrace E_i \right\rbrace_{i=1}^N$ is a covering of $S$
- I have freedom to position $E_i$ anywhere on the sphere.
I would like to know if there is:
- A minimal tiling of $\left\lbrace E_i \right\rbrace_{i=1}^N$ or a tiling with minimal overlap.
- An algorithm or way to solve a problem like this.
I imagine this problem is similar to the minimal tiling problem.