Is minimum distance between points on concentric ellipses constant? Given an ellipse, $E_1$ with radii $r_x, r_y$ I would like to know whether the minimum distance between any selected point, $P_a$, and $E_1$ is less than, say $D$.
I have seen the related question of finding the distance between a point and an ellipse. Rather than finding that minimum distance, I would like to solve it differently by creating an ellipse, $E_2$ with radii $r_x + D, r_y + D$ and then test whether the point $P_a$ falls within $E_2$, which is easy to do.
So, my question is, if the point $P_a$ falls within $E_2$, does it mean that the minimum distance between $E_1$ and $P_a$ is less than $D$?

Pictured below are


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*[Red] The (degenerate) ellipse $E_1$ with axes $1$ and $0$.

*[Green] The set of points exactly one unit from $E_1$.

*[Blue] The ellipse $E_{1+1}$ with axes $1+1$ and $0+1$.


All points within $E_{1+1}$ are within one unit of $E_1$ , but the converse is false. There are points within one unit of $E_1$ that don’t lie within $E_{1+1}$. 

Here’s a picture showing the “correct” (go out a distance $d$ perpendicular to the ellipse) and “approximate” (go out to an ellipse with axes larger by $d$) curves at two distances ($2$ and $6$) from one quarter of an original ellipse with axes $3$ and $1$. At least I hope it’s the right picture! As Matt suspected, the approximation is not bad at all.

