What do you mean by the "envelope" of the $\mathcal{E}_1$ set of ellipses?
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[![Ellipses][1]][1]
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<sup>
$\mathcal{E}_1$: $a=1$; $e=0,0.1,\ldots,0.9,1$.
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Added after OP change, now $\mathcal{E}$:
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[![Ellipses2][2]][2]
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<sup>
$a=2$; $e=0,0.1,\ldots,a$.
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  [1]: https://i.sstatic.net/yjrU7.jpg
  [2]: https://i.sstatic.net/d7uYS.jpg