are the envelopes of the following sets of ellipses "classical" curves of mathematics, that appear naturally as the solution of a mathematical/physical problem

\begin{aligned} \mathcal{E}_1 &:=\Biggr\{\quad\ \frac{x^2}{a^2}\quad+\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a) \\ \mathcal{E}_2&:=\Biggr\{\frac{(x-e)^2}{a^2} +\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a) \end{aligned}

are the envelopes of the following set of ellipses "classical" curves of mathematics, that appear naturally as the solution of a mathematical/physical problem;

$$\mathcal{E}:=\Biggr\{\frac{(x-e)^2}{a^2} +\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a)$$

the foci of these ellipses have coordinates $(0,0)$ and $(2e,0)$

are the envelopes of the following sets of ellipses "classical" curves of mathematics, that appear naturally as the solution of a mathematical/physical problem

\begin{aligned} \mathcal{E}_1 &:=\Biggr\{\quad\ \frac{x^2}{a^2}\quad+\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a) \\ \mathcal{E}_2&:=\Biggr\{\frac{(x-e)^2}{a^2} +\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a) \end{aligned}

The calculation of the envelope of the $\mathcal{E}_2$ requires the solution of a quintic polynomial and thus can't be an algebraic curve; that is why I am especially interested in the properties of that envelope.

are the envelopes of the following sets of ellipses "classical" curves of mathematics, that appear naturally as the solution of a mathematical/physical problem

\begin{aligned} \mathcal{E}_1 &:=\Biggr\{\quad\ \frac{x^2}{a^2}\quad+\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a) \\ \mathcal{E}_2&:=\Biggr\{\frac{(x-e)^2}{a^2} +\frac{y^2}{a^2-e^2}-1\ =\ 0\Biggr\},\quad e\in[0,a) \end{aligned}