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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.

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