Background and Motivation
Consider the following equation of family of ellipses in polar coordinates
$$ r(\theta, \alpha ) = a\;\frac{- e \cos(\theta+ \alpha )+ \sqrt{ 1 - e^2 \sin^2(\theta+\alpha)}}{(1 - e)(1 - e \cos \alpha)} $$
where $r$ and $\theta$ are usual polar coordinates and $e$ is the eccentricity of the ellipse. This equation describes a family of ellipses which are rotated around their focus by angle $\alpha$ such that $x= \dfrac{a}{1-e}$ is their envelope. See this post for the derivation of the equation by BLUE. The following animation given by BLUE helps to imagine better
The above set of expanding/contacting ellipses rotating about their focus was constructed by BLUE to have a single horizontal envelope . However, another envelope is found for this family of curves in this post, which was not envisaged at the outset, (not bargained for). The second blue line in the following picture shows the second envelope.
Question
Under what circumstances do we have an envelope for an envelope described by a single parameter/geometrical constant in such a set of smooth curves?
EDIT1:
Depicting the two parameter ($\epsilon$ and rotation) set of x-axis touching ellipses.
Second part of ellipse expansion after further upward expansion of major axis is omitted ,can be implemented later.
In the polar focal form there are two constants, $ \epsilon ,\, p $ the eccentricity and semi-latus rectum.
There are many ways of doing it.
1) Keep $p$ same, vary $\epsilon$.
2) Keep $\epsilon$ same,vary $p$.
3) Choose an arbitrary function connecting them.
Choice is for first option.
To touch a pre-fixed envelope Y = 5 the ellipse hinges about focus and droops by an angle:
$$ = \cos^{-1}\frac{Q}{2 e p Y}; \, Q^2 = 2 p^2 Y^2 (1+ e^2) - Y^2 (1-e)^2 $$
( polar coordinates) radius at tangential contact is given by:
$$ r_{T} = \frac{2 p Y^2}{p^2 +Y^2(1-e^2)} $$
Constants for above set of ellipses are $ (a =5, b=3, e =0.8, Y= 5.) $
It is however noted that the envelope appears as desired and that there is no envelope of envelope.
