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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:

Ellipse with Auxiliary Circle(s)

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?

Prime Mover
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  • Here a definition of the eccentric circles of the ellipse and how they can be used to construct the ellipse (scroll down): https://manasataramgini.wordpress.com/2018/07/06/the-hearts-and-the-intrinsic-cassinian-curve-of-an-ellipse/ One of them is in fact the auxiliary circle. – Intelligenti pauca Sep 26 '23 at 16:20
  • @Intelligentipauca which one? – Prime Mover Sep 26 '23 at 16:55
  • @MarianoSuárez-Álvarez But what are they in this context? That's my question. – Prime Mover Sep 26 '23 at 16:56
  • Did you read that page I linked? It's all explained very clearly. The auxiliary circle is the usual one, tangent at both vertces of the ellipse. The circles are eccentric between them, hence their name. – Intelligenti pauca Sep 26 '23 at 17:21
  • @Intelligentipauca The point is "auxiliary circle" was not mentioned anywhere on the page. – Prime Mover Sep 26 '23 at 17:30
  • True, but the definition given there of "eccentric circles of an ellipse" matches with the definition you found in the Penguin Dictionary of Mathematics. – Intelligenti pauca Sep 26 '23 at 18:22
  • @Intelligentipauca Yes I get that, all good and all, but it doesn't help me get to the answer of what was behind Nelson's thinking when he wrote that entry into that encyclopedia. – Prime Mover Sep 26 '23 at 19:06
  • Did you search for "eccentric circles of ellipse" inside you dictionary? – Intelligenti pauca Sep 26 '23 at 19:37
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    According to this other dictionary, the eccentric circles of an ellipse are those having as diameters the major and minor axis: https://archive.org/details/in.ernet.dli.2015.146873/page/n141/mode/1up – Intelligenti pauca Sep 26 '23 at 19:43
  • @Intelligentipauca Seriously misleading terminology that. Eccentric means off-centre, while these circles are definitely plumb dead centre of that ellipse. – Prime Mover Sep 26 '23 at 19:54
  • Indeed. But this is precisely the definition given by your Penguin Dictionary: “eccentric circle One of two circles that have the same centre as a central conic and diameters equal to the conic’s axes.” That settles the matter, I think. But you should have looked that up before asking here, in my opinion. – Intelligenti pauca Sep 26 '23 at 20:17
  • @Intelligentipauca That dictionary is so similar to the Penguin dictionary that I'm sure Nelson used it as the basis of his. But yes, you are right, I should have looked it up instead of posting. Perhaps everybody should look it up instead of posting. In fact, let's get rid of StackExchange altogether and tell everybody they should look everything up before posting. – Prime Mover Sep 26 '23 at 20:28

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