Context: Penguin Dictionary of Mathematics, 4th Edition (2008) ed. David Nelson.
I'm studying this as a course of self-study.
In the above dictionary we have this:
auxiliary circle One of the two eccentric circles of an ellipse or hyperbola. It is used in obtaining the parametric equations for the curve.
This is fundamentally at odds with every definition I've seen for the auxiliary circle, which appears to be the circle whose centre is at the centre of the conic section, which is tangent to the conic where it intersects its major axis (that is, its vertices).
By "eccentric circle", I believe (but cannot confirm) that an eccentric circle is a circle whose centre is at a focus and which is tangent to the conic at the nearer vertex to the focus.
And I have a vague suggestion in my mind that these two circles are called the associate circles of this conic section, but I cannot find anything corroborating my suspicion.
To clarify:
The ellipse $E$ in question is blue, with foci at $F_1$ and $F_2$.
I understand the auxiliary circle of $E$ is the circle $C_1$, in red.
However, I believe that Nelson is defining the auxiliary circles as being $C_2$ and $C_3$, which may or may not be called the "associate circles".
Everywhere I look the auxiliary circle is defined as in $C_1$.
I can find nothing suggesting that Nelson's definition is in any way correct.
Is it the case that this source work has a mistake, or is this indeed a variant definition that can be found somewhere in the literature?