What do you mean by the "envelope" of the $\mathcal{E}_1$ set of ellipses?
[![Ellipses][1]][1]
$\mathcal{E}_1$: $a=1$; $e=0,0.1,\ldots,0.9,1$.
Added after OP change, now $\mathcal{E}$:
[![Ellipses2][2]][2]
$a=2$; $e=0,0.1,\ldots,a$.
[![Ellipses][1]][1]
$\mathcal{E}_1$: $a=1$; $e=0,0.1,\ldots,0.9,1$.
[![Ellipses2][2]][2]
$a=2$; $e=0,0.1,\ldots,a$.
Joseph O'Rourke
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